Learn Calculus (Best Online Course) - Solved examples

Use substitution to solve the integral, with t = x e^x and dt = e^x(x+1) dx, converting sec^2 x to tan t and yielding tan(x e^x).

Apply substitution to the integral of e to the arcsin x over the square root of one minus x squared, with t = arcsin x, giving e^t + C.

Practice integration by substitution with sin x and cos x. Substitute t = cos x or t = sin x to transform integrals into polynomials and get -cos 2x/3.

Explore integration by substitution with t = sine inverse x and t = log ten x/2, yielding -1 / sine inverse x and (log ten x/2)^2 / 2.

Learn substitution in integrals by setting t = 10^x to get (10^x)^2/2 + C, and t = sin x to obtain sin^5 x/5 + C.

Learn to evaluate the integral of sin^2 x dx using sin^2 x = (1 − cos 2x)/2, split the integral, and obtain sin(2x)/4 + c.

Apply trig identities to solve integrals: rewrite sin^3 x via sin 3x and integrate. Transform cos 5x cos 3x using 2 cos a cos b, and integrate to sine terms.

Learn to integrate powers of trigonometric functions, specifically sin^5 x and cos^7 x, using substitutions to convert to polynomials and perform the integrations.

Apply the identity cos^2(x/2) = (1 + cos x)/2 to rewrite the integrand and integrate term by term to obtain x/2 + (1/2) sin x + C.

Apply integration by partial fractions to rational functions with linear factors. Resolve into a/(x+1) + b/(x+2), determine a and b, and integrate to log(x+1) - log(x+2).

Decompose the function x+1 over x(x^2-4) into partial fractions and identify constants a, b, and c. Integrate the resulting terms to obtain logarithmic expressions with coefficients -1/4, -1/8, and 3/8.

Apply substitution with t = 10x and partial fraction decomposition to reduce the integral to logarithms, yielding log((10x+1)/(10x+2)) plus a constant.

Learn integration by partial fractions through substitutions, t = sin x and t = x^2, decomposing into partial fractions to express results as logarithms of sin x and x.

Explore integration by parts using the x sin x example and the liate order to choose u and dv, yielding sin x minus x cos x.

Explain integration by parts through the example ∫ x e^x dx, with u = x and dv = e^x dx, yielding x e^x - e^x + c.

Explore substitution methods to evaluate integrals, including ∫ x(x^2+1)^4 dx by setting t=x^2+1 with dt=2x dx, yielding (x^2+1)^5/10 + C, and ∫ x^{n-1} cos(x^n) dx with t=x^n, giving (1/n) sin(x^n) + C.

this demonstrates integration by parts, using u = x and dv = sin 2x, applying ∫ u dv = uv − ∫ v du to compute ∫ x sin 2x.

Apply integration by parts to compute integrals of x cos 2x and x e raised to two x by choosing u and dv and using uv minus ∫ v du.

Learn integration by parts with t = sqrt x to solve sin sqrt x and cos sqrt x integrals, yielding 2(√x sin √x + cos √x) + c.

Apply integration by parts to x log x and x^100 log x, choosing u as log x and dv as x or x^100, using power rule x^n dx = x^{n+1}/(n+1).

Compute the integral of cos(log x) dx using substitution t = log x and integration by parts, yielding x/2 [cos(log x) + sin(log x)] + C.

Master definite integration with solved examples on using substitution in ∫ dx/(2x+3) to obtain (1/2) log(11/7), and evaluating ∫ e^(2x) and ∫ sin(3θ) from 0 to limits.

Explore solved definite integrals by applying substitution and recognizing derivatives in the integrand. Apply sine double-angle, logarithmic, and exponential rules to compute results across examples from the video.

use the definite integration property with x replaced by a plus b minus x to convert to sine and cosine forms via nth roots, then sum equations to get integral.

Use the symmetry property ∫0^a f(x) dx = ∫0^a f(a - x) dx to simplify definite integrals, illustrated by results like a/2 and pi/4.

Applies the symmetry trick x -> (lower limit + upper limit) - x to definite integrals, turning complex cube-root and root expressions into simple 1/x integrals and solving.

Find the area between y = x and y = x^2 by integrating from 0 to 1, with y1 = x and y2 = x^2, yielding 1/6 square units.

Find the area between the parabola y = x^2 + 1 and the line y = 2x + 1 by integrating from 0 to 2 to obtain 4/3 square units.

Compute area under the curve y = x^3 - 5x^2 + 4x from x = 0 to 3 using integration, splitting at x = 1 into areas a1 and a2.

Compute ellipse area by definite integration in the first quadrant for x^2/25 + y^2/9 = 1, then apply the same method to circle x^2 + y^2 = 36.

determine the area under the curve bounded by the parabola y = 4x - x^2 and the x-axis using a definite integral from 0 to 4, yielding 32/3.

Learn to compute the area between parabolas y^2=9x and x^2=9y using GeoGebra and single-variable integration, finding intersections and evaluating the integral to obtain 27 square units.

Master the product rule for derivatives with two- and three-function cases, using examples like x e^x, x log x, and x^2 5^x cos x.

Learn the quotient rule for derivatives, using dy/dx = (v du/dx − u dv/dx)/v^2, with examples like (x+1)/(x−1) and sin x/(1−cos x) to illustrate step-by-step differentiation.

Apply the chain rule to derivatives of composite functions like (x^2+1)^10 and e^(3x) with inner derivatives, and explore product and trig derivatives such as sin(3x+2) and cos^2 x.

Explore the chain rule with diverse derivative examples, including arcsin x and its 1 over root(1−x²), and derivatives of √x, √(sin 2x), and log-based expressions.

Explore the chain rule with direct method to differentiate dy/dx for functions raised to n, inner substitutions, and exponential, sine, and square-root examples.

Explore derivatives of inverse trigonometric functions via substitutions, simplifying expressions such as arcsin and arccos of composite forms, and deriving concise formulas.

Learn to differentiate implicit functions using logarithmic differentiation, quotient and product rules, and chain rule, with concrete examples such as dy/dx expressions for x^y = e^{x−y} and e^y = y^x.

Differentiate one function with respect to another using chain and product rules. Compute dy/dz = (dy/dx)/(dz/dx) in examples like log x sin x, arctan(4x) and arcsin x.

Explore the slope of a curve by defining tangent and normal lines, derive slope as dy/dx, and show that the normal slope equals minus one over the tangent slope.

Apply derivatives to find tangent and normal lines, locate points where dy/dx=0 and where dy/dx=1 on parabolas y=x^2-6x+8 and y=7x-3x^2, yielding (3,-1) and (1,4).

determine the tangent to the curve y=9x^2-12x+7 parallel to the x-axis by differentiating, solving 18x-12=0 for x=2/3, finding y=3, and writing y=3 as the tangent.

learn to compute curve slopes using derivatives, including dy/dx for y = cube root of x with chain rule, and parametric case x = cos 2θ, y = sin^3 θ.

Derive acceleration from velocity as a function of displacement, showing that a = -k v^2 for v = e^{-k s}; relate v^2 to s e^{s} to complete the expression.

Compute dy/dx for x=1/t, y=t-1/t at t=2 to find tangent slope -5, then the normal slope 1/5 yields the normal equation x - 5y + 7 = 0.

Compute the stationary speed by differentiating the rate function, set dy/dv to zero to find v = 20, and verify a minimum with the positive second derivative.

Differentiate y = 120 t - 3600 t^2 to obtain y', set it to zero for t = 1/60 s, and confirm a maximum depth of 1 meter.

Explore velocity as the rate of change of displacement and acceleration as the rate of change of velocity, using first and second derivatives of displacement with respect to time.