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Calculus Integration Insanity with the Math Sorcerer

Name: Calculus Integration Insanity with the Math Sorcerer
Rating: 4.9 (167 reviews)

****250+ Videos**** This is the ULTIMATE Integration Course!!

Created byThe Math Sorcerer

Last updated 5/2026

English

English [Auto],Korean [Auto],

What you'll learn

How to Integrate using Techniques Taught in Calculus 1
How to Integrate using Techniques Taught in Calculus 2
How to Integrate using Integration by Parts
How to Integrate using u-substitution
How to Integrate Basic Trigonometric Functions
How to Integrate using Tabular Integration
How to Integrate Triognometric Integrals involving Powers of Sine and Cosine
How to Integrate Trigonometric Integrals involving Powers of Secant and Tangent
How to Use Trigonometric Substitution to Integrate
How to Integrate with Bases other than e
How to Integrate using the Power Rule
How to Integrate with Partial Fractions
How to Integrate Hyperbolic Functions
How to Evaluate Definite Integrals
How to Compute Integrals that Lead to the Inverse Trigonometric Functions
How to Compute Integrals Involving e^x
How to Compute Integrals with Formulas involving the Integral of 1/x
How to Compute Integrals using Various Trigonometric Identities
How to Integrate using Various Techniques

Course content

26 sections • 254 lectures • 14h 4m total length

The Theory Behind Integration (Requires Knowledge of Derivatives)10:41
Explore the theory of integration by defining antiderivatives, such as x^2 and x^2+c; learn indefinite integration, the constant of integration, and integral notation.
The Basic Integration Rules7:02
Master the basic integration rules: integrate zero to a constant, constants as kx + c, apply the power rule for x^n (n ≠ -1), and use linearity to handle sums.
Formulas for the Integrals of Trigonometric Functions4:49
Explore core trig integration formulas, including ∫ cos x dx = sin x + C and ∫ sin x dx = -cos x + C, with checks by differentiation.
Example 11:08
Apply the power rule to integrate x to the minus four, rewrite as -1/3 x to the minus three, include +C, yielding -1/3 x to the minus three.
Example 21:14
Learn to integrate a linear combination of cosine and sine by reversing differentiation, yielding 3 sin x minus 8 cos x plus the constant of integration.
Example 32:13
Learn to integrate without u-substitution by rewriting sqrt(5x) as sqrt(5) sqrt(x), pull out constants, apply the power rule to x^(-1/2), and obtain 2 sqrt(x)/sqrt(5) + C.
Example 42:26
Evaluate the definite integral by rewriting the integrand to powers of x and applying the power rule to x^2 and x^-2, yielding x^3/3 - (1/16)(1/x) + C.
Example 51:23
Master the integral of sec^2 x by recognizing it as the derivative of tan x, yielding tan x plus a constant.
Example 62:06
Apply the power rule to integrate terms with exponents, raise each exponent by one, multiply by the reciprocal, and obtain the final result 2/3 x^(3/2) + 1/4 x^(1/2) + C.
Example 71:12
Learn the power rule for integration by adding one to the exponents and dividing by the new exponent, include the constant of integration, and verify by differentiating.
Example 81:43
Apply the power rule to integrate one over the square root of x, rewriting as x^(1/2) and obtaining 2 sqrt(x) + C.
Example 91:26
Break the integral into t^(1/2) minus t^(5/2); apply the power rule to obtain (2/3) t^(3/2) minus (2/7) t^(7/2) plus C.
Example 101:03
Learn to apply the power rule for integration by adding one to the exponent and dividing by the new exponent, with a note on the negative one exception.

Example 11:59
Master integration by breaking the function into simpler powers, applying the power rule, and using reciprocals to divide by the new exponent, with a final plus C.
Example 21:23
Learn the power rule for integration: add one to the exponent, divide by it, use the reciprocal for easier division, include the constant of integration, and verify with differentiation.
Example 31:13
Expand the product using foil to turn (x+2)(x-3) into x^2 - x - 6. Then apply the power rule to integrate each term and add the constant of integration.
Example 40:53
Learn to integrate sine x by recognizing that the derivative of negative cosine x is sine x, yielding negative cosine x plus a constant, verified by differentiation.
Example 51:08
Practice integrating a trig expression by matching derivatives to antiderivatives. Use cosine's negative derivative, keep constants, and verify your result by differentiating to check.
Example 62:02
Expand the product (x+1)(3x-2), then apply the power rule to integrate the resulting polynomial, yielding x^3 + x^2/2 - 2x + C.
Example 71:01
Think backwards when integrating trig functions: identify a function whose derivative matches the integrand, using tangent relationships as clues to quickly find the antiderivative.
Example 80:40
Explore integrating expressions, apply the power rule, and note that the derivative of tangent is second squared, so integrating second squared yields tangent plus C, though not the final answer.
Example 91:20
Use 1 minus cos squared x equals sin squared x to rewrite cos x over sin^2 x as cot x csc x, giving -csc x + C.
Example 101:41
Apply the identity sin two x equals two sin x cos x to simplify the integrand. Integrate to obtain two sin x plus C.

Example 12:26
Master indefinite integrals using u-substitution with u = x^2 + 17, align coefficients to match the integral, and apply the power rule to obtain the antiderivative plus a constant.
Example 22:30
Use u-substitution with u = x^2 - 12x + 2; du = (2x - 12) dx; integrate to sqrt(u) + C, giving sqrt(x^2 - 12x + 2) + C.
Example 32:10
Use a u-substitution with u = sin(3x) to transform the integral; apply the chain rule and divide by 3, yielding the result (1/3) tan(sin(3x)) + C.
Example 42:09
Use a u-substitution with u = 17 + 1/x; since du = -1/x^2 dx, rewrite the integral as -∫u^6 du and obtain -u^7/7 + C, then back-substitute.
Example 52:05
Solve an integral by substitution, setting u = 3 + cos x, applying the power rule, and obtaining the antiderivative with a constant of integration.
Example 61:47
Apply u-substitution with u = tan x to integrate the cube root of tan x times sec^2 x, yielding (3/4) tan^{4/3} x + C.
Example 71:40
Use u-substitution to evaluate the indefinite integral of x^2 sin(x^3), substitute u=x^3, obtain -(1/3) cos(x^3) + C, and verify by differentiation.
Example 82:07
learn to use u-substitution to integrate sin(2x), pull out constants, and obtain the antiderivative -(1/2) cos(2x) + C, with clear substitution steps.
Example 93:49
Apply u-substitution with u equals x plus 2, rewrite x minus 5 as u minus 7, and use power rule to obtain the antiderivative in terms of x plus 2.
Example 103:00
Use substitution with u = tan(4x) to handle the square root, pull out constants, and obtain the integral (1/6) [tan(4x)]^(3/2) + C.

Example 14:03
learn to integrate the indeterminate integral (2x+1)/sqrt(x+4) using u-substitution, solve for x in terms of u, and obtain (4/3)(x+4)^{3/2} - 14(x+4)^{1/2} + C.
Example 23:40
Multiply by the conjugate to apply the difference of squares, rewriting 1/(cos x − 1) as (cos x + 1)/(cos^2 x − 1); then integrate using cotangent and cosecant squared.
Example 32:58
Multiply by (1+sin x)/(1+sin x) to apply the difference of squares, rewrite the integrand as sec^2 x plus tan x sec x, and obtain tan x + sec x.
Example 43:18
Demonstrates solving 1/(1−cos x) by multiplying by (1+cos x)/(1+cos x), applying difference of squares to obtain (1+cos x)/sin^2 x, then rewriting with familiar trig functions.
Example 52:35
Use u-substitution with u = tan x to turn the integral into a power of u, then integrate to obtain one-half tan squared x plus C.
Example 62:18
Use u-substitution with u = 1 + 1/x, derive du = -1/x^2 dx, transform the integral to ∫ u^4 du, then back-substitute to get -(1/5)(1 + 1/x)^5 + C.
Example 71:49
Use a u-substitution with u equals x cubed plus seven, convert du to 3x^2 dx, rewrite the integrand as u^(-1/2), and obtain 6 sqrt(u) plus 7x^3 plus 7 plus C.
Example 81:55
Practice integrating a power function by substitution: set u = 2 − 3x, divide by −3, and apply the power rule to obtain −(2−3x)^{11}/33 + C.
Example 92:25
Learn to integrate cosine of 2x using a u-substitution: set u = 2x, factor out 1/2, and obtain (1/2) sin(2x) + C.
Example 102:11
Solve an indefinite integral using u-substitution with u = 1/x, rewrite as x^-1, apply the power rule, transform to the integral of sine u du, and obtain cos(1/x) + C.

Introduction to the Formulas Needed5:49
Learn common integration formulas, including ∫1/x dx = ln|x| + C, apply u-substitution for expressions like 2−x, and explore tan, cot, and sec integrals.
Example 12:23
Use u-substitution with u = 5 - 6x to integrate 1/(5-6x) dx. Pull out the constant and obtain the result -1/6 ln|5-6x| + C.
Example 21:54
Apply substitution with u = ln x to convert the integral of ln(x^3)/x into ∫u^3 du, then use the power rule to obtain (ln x)^4/4 + C.
Example 32:33
Apply u-substitution to integrate the given function, choosing u = 6 + x^(1/3) and transforming the differential to reveal a logarithmic antiderivative.
Example 42:23
Example 4 shows using u-substitution with u = 2x, extracting a 1/2 factor, applying a negative natural log of absolute value plus a tangent term, then back-substituting to x.
Example 52:42
The lecture demonstrates integrating x over three using a u-substitution, applying the secant integral formula ln|sec x + tan x|, and obtaining 3 ln|sec(x/3) + tan(x/3)| + C.
Example 61:36
Apply u-substitution by letting u be the bottom piece; use du as the derivative and recognize the derivative of sine is cosine, convert the integral to ln|u|+C, then substitute back.

Example 11:11
Example 21:43
Master substitution to integrate five times the cosine of x^2. Let u = x^2, adjust coefficients, and obtain (5/2) sin(x^2) + C.
Example 32:04
Use u-substitution with u = cos x to rewrite the integral, manage the negative sign via du = -sin x dx, then integrate cos u and include the constant C.
Example 42:30
Apply u-substitution by setting u = 1 − x^2 and use the power rule to integrate x sqrt(1 − x^2), yielding -(1/3)(1 − x^2)^{3/2} + C.
Example 52:52
Solve an indefinite integral by substituting u = cot(3x) and using the chain rule, then apply the power rule to integrate a square root, yielding a cot(3x) expression plus C.
Example 64:58
Use substitution with u = x^2 - 1 to transform the integral x^3 sqrt(x^2 - 1) dx. Apply the power rule to obtain the final expression.
Example 72:00
Use a u-substitution with u = cos x to integrate sin x times cosine of cosine x, giving -sin(cos x) + C.
Example 83:41
Use u-substitution with u = x − 1 to transform the integral, then apply the power rule to obtain (2/3)(x − 1)^(3/2) + 2√(x − 1) + C.
Example 92:00
Apply a u-substitution with u=3x and pull out 1/3 to use the secant antiderivative ln|sec u + tan u|, yielding (1/3) ln|sec(3x) + tan(3x)| + C.
Example 102:10
Apply u-substitution to integrate three times Kosygin squared of seven x squared, yielding an antiderivative -3/14 cot(7x^2) + C.

Example 11:41
Learn to integrate e^(3x-6) using a u-substitution, setting u = 3x - 6 and pulling out a 1/3 factor. Conclude with (1/3) e^(3x-6) + C and recover the original variable.
Example 22:25
Use a u-substitution with inside piece 2 - x to integrate e^x times sqrt(2 - x), rewrite the root as u^(1/2), apply the power rule, and include C.
Example 33:29
Break up the two-term numerator over the single-term denominator and apply exponent rules to subtract exponents when dividing. Use the shortcut of dividing by the coefficient to integrate exponentials.
Example 41:10
Apply u-substitution to integrate e^{b x} with respect to x, pulling out constants to get (1/b) e^{b x} + C, as shown by the example of e^{2x} divided by 2.
Example 51:20
Use substitution to evaluate an indefinite integral involving e^{x^3}. Derive du from 3x^2 dx, divide by three, substitute, integrate in u, then back-substitute to x and include constant of integration.
Example 62:07
Use a u-substitution with u = x − b to transform the integral into a ln|u| + C, showing partial fractions are unnecessary after cancellation.
Example 72:15
Rewrite the integral to reveal e^x in the numerator, then substitute u = e^x + 1 to transform the integral into du/u and obtain 2013 ln|e^x+1| + C.
Example 82:42
Learn to evaluate the integral ∫ x/(1+e^x) dx by a clever algebraic trick and a u-substitution: set u = 1+e^x, yielding the antiderivative x − ln|1+e^x| + C.
Example 92:43
Explore integrating x over 1+ e^x dx using algebraic tricks, split into two integrals, and apply u-substitution u=1+ e^x to obtain a log form with plus C.
Example 101:40
this example solves the integral of x e^{x^2} by letting u = x^2 and du = 2x dx, yielding 1/2 e^{x^2} + C.

Example 11:47
Apply substitution u = 1 + cot x to transform the integral, then obtain -ln|u| + C with u = 1 + cot x.
Example 21:33
Apply a u-substitution with u = 1 + sec theta to evaluate the indefinite integral of sec theta tan theta over 1 + sec theta, yielding ln|u| + C.
Example 31:23
Apply a u-substitution with u = ln x to convert the integral into ∫ sin u du, and obtain -cos(ln x) + C.
Example 42:18
In this example, evaluate the indefinite integral sin^2 x over 1+cos^2 x by applying chain rule and recognizing the du/u form to obtain a negative natural log of 1+cos^2 x.
Example 50:57
Learn to integrate cot x by substitution u = sin x, since cot x = cos x over sin x, yielding the natural log of |sin x| plus C.
Example 63:54
Use synthetic division to divide x^4 by x-1, obtaining quotient x^3+x^2+x+1 and remainder 1, then integrate to x^4/4 + x^3/3 + x^2/2 + x + ln|x-1| + c.
Example 71:32
Apply a substitution with u = cos x and du = -sin x dx to integrate e^{cos x} sin x dx, yielding -e^{cos x} + C.
Example 81:53
Substitute u = x + B; the integral becomes (1/A) ∫ 1/u du = (1/A) ln|u| + C, yielding (1/A) ln|x + B| + C.
Example 92:26
Apply u-substitution with u = x - e^(-x) to simplify the integral of (x + e^(-x))/(x - e^(-x)) dx, then obtain ln|u| + C.
Example 102:41
Use u-substitution with u = ln x to transform the integral of sqrt(u) du, apply the power rule, and obtain (2/3)(ln x)^(3/2) + C.

Formulas7:04
Learn key exponential and logarithmic formulas, including d/dx(a^x)=a^x ln a, d/dx(log_a x)=1/(x ln a), and ∫a^x dx=a^x/ln a + C, with change-of-base ideas.
Example 11:38
Apply the power rule to integrate x^8 and use the exponential rule for 5^{-x}. Rewrite 5^{-x} as 1/5^x and obtain x^9/9 + 5^{-x}/ln(1/5) + C.
Example 21:14
Use a u-substitution with u = sin x to integrate 8^{sin x} cos x dx. Result equals 8^{sin x}/ln 8 + C after substituting back.
Example 31:14
Apply u-substitution to integrate 2^{sin x} cos x with respect to x. Let u = sin x, then integrate 2^u du to obtain 2^{sin x}/ln 2 + C.
Example 42:14
Apply u-substitution with u = 3x to handle the cosine term and the inside derivative, yielding a 1/3 factor. Integrate 2^u and divide by the natural log of 2.
Example 51:58
Rewrite 2^t * 5^t as 10^t using exponent rules, then integrate 2^t and 10^t with ∫ a^x dx = a^x/ln a, yielding 2^t/ln2 + 10^t/ln10 + C.

Formulas8:29
Learn essential integration formulas for 1/√(a^2−x^2) and 1/(a^2+x^2), with memorized arcsin and arctan forms. Apply substitutions and simple examples to see these patterns.
Example 11:17
Apply the standard formula for integrals of 1/(a^2+x^2) to obtain arctan(x/a), and verify by noting the derivative of arctan is 1/(1+x^2).
Example 21:22
Use the standard arcsin formula with a=1 to evaluate the integral 1/sqrt(1 - x^2). Recognize that the derivative of arcsin is 1/sqrt(1 - x^2) and apply it to confirm the result.
Example 31:47
learn to recognize the standard integral form 1/(a^2+x^2), rewrite the integrand to fit the formula, and apply the arctan result (1/a) arctan(x/a) + c.
Example 42:32
Split the integral into two parts, apply a u-substitution for the first and arcsin for the second, yielding -sqrt(1 - x^2) + 3 arcsin(x) + c.
Example 55:04
Apply u substitution and arc sine to evaluate an indefinite integral, rewriting x+5 as (x-4)+9 to isolate a term, and then integrate the resulting expressions.
Example 62:36
Rewrite the integral to match the arcsin formula, substitute u = 5t^2, pull out 1/2, and obtain (1/2) arcsin(5t^2) + C.
Example 74:50
Learn to split an indefinite integral into an ln term and an arctan term using a u-substitution with x plus two, yielding (1/2) ln((x+2)^2+16) - (3/2) arctan((x+2)/4) + c.
Example 82:35
Tackles an indefinite integral by recognizing an arctan form, applying the substitution u = e^{2x}, and deriving the antiderivative (1/18) arctan(e^{2x}/9) + C.
Example 92:23
Rewrite the integral as an arctan form, substitute u = T^2, pull out constants, and apply the arctan formula to obtain (1/6) arctan(T^2/3) + C.
Example 101:48
Let u equal the natural log of x and rewrite the integral. Apply the arc sine formula to arrive at ln(x)/4 + C.

Requirements

Knowledge of Differential Calculus (You will Need to Know What a Derivative is)

Description

This is literally the ULTIMATE Course on Integration!!

The most important requirement is that you know what a derivative is. If you know that then you can jump into this course as it starts from the very basics of integration.

Basically just,

1) Watch the videos, and try to follow along with a pencil and paper, take notes!

2) Try to do the problems before I do them(if you can!)

3) Repeat!

Integration is an absolutely beautiful subject. I hope you enjoy watching these videos and working through these problems as much as I have:)

Note this course has lots of very short videos. If you are trying to learn math then this format can be good because you don't have to spend tons of time on the course every day. Even if you can only spend time doing 1 video a day, that is honestly better than not doing any mathematics. You can learn a lot and because there are so many videos you could do 1 video a day for a very long time. Remember that math can be challenging and time consuming, so if you just do a little bit every day it can make your journey much more enjoyable. I hope you enjoy this course and learn lots of mathematics.

Who this course is for:

Anyone Taking Calculus 1
Anyone Taking Calculus 2
Anyone Who Wants to Learn How to Integrate
Anyone Who Wants to Refresh Their Integration Skills

Calculus Integration Insanity with the Math Sorcerer

What you'll learn

Explore related topics

Course content

Level 1: [New Concept] Basic Integration13 lectures • 38min

Level 2: [Practice Problems] Let's Do These Fun Integrals:)10 lectures • 13min

Level 3: [New Concept] Integration by Substitution10 lectures • 24min

Level 4: [Practice Problems] A little harder this time:)10 lectures • 27min

Level 5: [New Concept] Formulas for 1/x, sec(x), csc(x), tan(x), cot(x)7 lectures • 19min

Level 6: [Practice Problems] Even More Problems!10 lectures • 25min

Level 7: [New Concept] The Integral of e^x is e^x, let's do this!10 lectures • 22min

Level 8: [Practice Problems] Yes More Integrals:)10 lectures • 20min

Level 9: [New Concept] Integral of a^x6 lectures • 15min

Level 10: [New Concept] Integrals Involving Arcsine and Arctangent11 lectures • 35min

Requirements

Description

Who this course is for: