Calculus 2 with the Math Sorcerer

Learn two essential integration formulas leading to inverse trig functions: ∫ dx/√(a^2−x^2)=arcsin(x/a) and ∫ dx/(a^2+x^2)=(1/a) arctan(x/a). See substitutions and examples.

Apply the standard formula for integrating 1/(a^2+x^2) to derive (1/a) arctan(x/a) + c, and recognize that the derivative of arctan is 1/(1+x^2).

Compute the integral of 1 over sqrt(1 minus x squared) using the standard formula to get arcsin(x) plus a constant, or confirm it via trig substitution and arcsin's derivative.

Identify the form a^2 + x^2 and apply the 1/a arctan(x/a) formula to find the indefinite integral of 1/(a^2 + x^2), plus C.

Split the integral into x/√(1−x²) and 3/√(1−x²). Use u-substitution for the first and arcsin for the second to obtain -√(1−x²) + 3 arcsin x + C.

Use the standard formula ∫ dx / sqrt(a^2 − x^2) to evaluate ∫ dx / sqrt(13 − x^2), yielding arcsin(x/√13) + C.

Rewrite the integral 25/(1+25x^2) as 25/(1+(5x)^2) and substitute u=5x. This yields 5 arctan(5x) + C, using the arctangent form.

Solve an indefinite integral that resembles arcsin by applying a substitution to rewrite the integrand as 1/√(1−u^2) and then use the arcsin formula with a constant.

Learn to integrate using the arctan formula by substituting u = cos x, handling a negative du, and obtaining -1/√6 arctan(cos x/√6) + C.

Tackle an indefinite integral by rewriting x+5 as (x-4)+9 to enable a u-substitution, yielding the arc sine for the second integral.

Decompose an indefinite integral into a log and an arctan part using a u-substitution and the arctan formula, yielding a half natural log term and a scaled arctan term.

Identify the indefinite integral as an arctan form by using the substitution u = e^{2x}. Obtain the result (1/18) arctan(e^{2x}/9) + C.

Apply a substitution u = tan x to convert the integral into an arcsin form, using du = sec^2 x dx and sqrt(1 − u^2). Conclude with arcsin(tan x) + C.

This lecture shows how to convert an indefinite integral to arctan form by rewriting the integrand and applying the arctan formula, with a substitution and constants pulled out.

Use u-substitution with u = ln x to transform the integral into an arcsin form, yielding 4 arcsin((ln x)/4) + C.

Complete the square on -x^2-4x to get a perfect square, substitute u = x+2, and apply the arcsin formula to evaluate the integral.

Rewrite x^4 as (x^2)^2, then use u-substitution with u = x^2 to simplify the integral. Obtain an antiderivative involving arctan and a constant of integration.

This lecture demonstrates solving an integral with a square root of 100 minus x squared using a u-substitution and the standard form for arcsin, yielding arcsin(x/10) plus C.

Explore evaluating the integral of (x+2)/(x^2+2x+5) by a u-substitution, completing the square to transform the denominator, and applying the arctan formula to finish.

Apply the formula for the indefinite integral of x over sqrt(a^2 - x^2) using trig substitution, yielding arcsin(x/a) plus C with a = sqrt(22). Emphasize pattern recognition.

Evaluate an indefinite integral by recognizing the standard formula for 1/(a^2+x^2) dx, use a u-substitution with u=5x after pulling out the 25, and obtain 5 arctan(5x) + C.

Students learn to solve a complex integral by completing the square, rewriting the integrand to the arcsine form, and using a u-substitution to yield 1/2 arcsin((x^2-1)/6) + C.

Apply the arctan integration formula to evaluate the given indefinite integral using a u-substitution. Identify a = 3 and express the antiderivative as -4/3 arctan((2 − x)/3) + C.

Apply the standard antiderivative for ∫ dx / sqrt(a^2 - x^2) to obtain arcsin(x/a) with a=10. Use a u-substitution to rewrite the integral and arrive at arcsin(x/10) + C.

Learn to complete the square on the denominator to form a perfect square trinomial, use a u-substitution with u = x^2+5, and apply the arctan formula to solve the integral.

Evaluate an indefinite integral by splitting the integrand, applying a substitution with 1 minus x squared, and using the arcsin formula to obtain the antiderivative plus a constant.

Solve a differential equation by finding the general solution, then apply the initial condition to obtain the particular solution. Use the arcsin formula with a=2 to get Y(x)=arcsin(x/2)+π, satisfying Y(0)=π.

Compute the derivative of the inverse sine without the formula, using the chain rule and a right-triangle approach to obtain 1 over the square root of 1 minus x squared.

Explore the full derivation of the derivative of arctangent, using inverse functions, the chain rule, and a right-triangle approach to obtain dy/dx = cos^2 y = 1/(1+x^2).

Differentiate implicitly with product and chain rules, use the arcsin derivative, evaluate at (0,0) to find slope -1, and obtain the tangent line y = -x.

Compute the tangent line to the implicit curve at (1,0) using implicit differentiation, find dy/dx, substitute the point, and apply the point-slope form to obtain y = -x + 1.

Differentiate the equation with respect to x using the product rule and arctan derivative to get dy/dx. Evaluate at (-4, 1) and apply point-slope form to find the tangent line.

Apply the first derivative test to f(x) = arctan x − arctan(x−5) to locate the critical point at x = 2.5, giving a maximum value f(2.5) ≈ 2.381.

Learn the definitions of the hyperbolic functions—sinh, cosh, tanh—and their reciprocals csch, sech, coth, with sample evaluations like sinh(0)=0 and sinh(ln2)=3/4, and preview derivatives, integrals, and identities.

learn how to evaluate hyperbolic functions such as sinh, cosh, and tanh using the formulas (e^x - e^{-x})/2 and (e^x + e^{-x})/2, with calculator tips.

Derive the hyperbolic identity cosh^2 x minus sinh^2 x equals 1 by expanding cosh x and sinh x as (e^x + e^{-x})/2 and (e^x - e^{-x})/2, then simplifying.

Proves a hyperbolic identity by expanding the right-hand side with sinh and cosh definitions and simplifying to sinh(x+y).

Learn to prove hyperbolic identities by deriving relation hyperbolic tangent squared plus hyperbolic secant squared equals one from cosh squared minus sinh squared equals one, via division by cosh squared.

Demonstrate how to prove a hyperbolic identity by applying the addition formula to sinh(x+y) and rewriting 2x as x+x, showing the left side equals the right side.

Learn the derivatives and integrals of hyperbolic functions, including sinh, cosh, and tanh, and apply chain rule and substitution to compute their antiderivatives.

Apply the chain rule to differentiate sin(9x): take the outer derivative cos evaluated at 9x and multiply by the inner derivative 9, giving 9 cos(9x).

Differentiate sech(5x^2) with the chain rule, yielding y' = -10x sech(5x^2) tanh(5x^2) and illustrating the inner function 5x^2.

Compute the derivative of tanh(4x^2+3x) using the chain rule: d/dx tanh(u)=sech^2(u) u', giving (8x+3) sech^2(4x^2+3x).

Differentiate g(x) = e^{cos x} using the chain rule; the derivative is -sin x e^{cos x}.

Differentiate tanh(3x^2) using the chain rule; apply the outer derivative sech^2(3x^2) and multiply by the inner derivative 6x.

Apply the product rule to differentiate (t/6) sinh(-3t) and use the chain rule on sinh(-3t), yielding the final result -(1/2) t cosh(-3t).

Use the chain rule to differentiate sin(5x); treat 5x as the inner function, whose derivative is 5, and bring the factor to the front for the final result.

Derive the natural log of a function with the chain rule, using one over the inside times its derivative; result: inside' over inside or tanh(x).

Differentiate (sech(5x))^2 using the chain rule, treating the outer power and inner function, and obtain -10 sech^2(5x) tanh(5x) as the final result.

Apply the chain rule to differentiate a function, rewriting with natural log to simplify, and derive a hyperbolic secant squared form of the derivative.

Compute the derivative of tanh(2x^2-7) by applying the chain rule and the formula tanh'(u)=sech^2(u), with u=2x^2-7, yielding 4x sech^2(2x^2-7).

Use the product rule and chain rule to differentiate x^2 plus three times the hyperbolic tangent of x/4, rewriting tanh(x/4) to simplify the derivative.

Compute the derivative of a function by differentiating a constant and the hyperbolic tangent, noting that the derivative equals negative hyperbolic cosh squared of t.

Apply the chain rule to differentiate cosh(x-9); multiply the derivative of the outer function by the derivative of the inside (1) to obtain sinh(x-9) as the final result.

Compute the derivative of f(x) = e^{sin x}, evaluate at x = 0 for slope 1, then apply the point-slope form to obtain y = x + 1.

Derive the general antiderivative of x cosh x by rewriting cosh x as (e^x + e^{-x})/2, applying substitution u=2x, and obtaining 1/4 e^{2x} + 1/2 x + C.

Practice solving an integral with hyperbolic functions using a u-substitution: sech x and tanh x, apply the power rule, and obtain -sech^5(x)/5 + c.

Compute the integral of sech^2(3x-2) by substitution u=3x-2, yielding (1/3) tanh(3x-2) and demonstrating how to handle the derivative 3.

In this example, apply substitution to simplify an integral with square roots, rewrite expressions, and pull out factors as you work toward the antiderivative.

Perform a u-substitution with u = cosh x to integrate sinh x / cosh^3 x dx, yielding -1/(2 cosh^2 x) + C.

We integrate cosh(4x) using a u-substitution with u = 4x, pull out the 1/4 factor, and obtain (1/4) sinh(4x) + C.

Demonstrate using a u-substitution to integrate nine minus eight x, letting u = 9 − 8x, pulling out a constant, and obtaining −1/8 cos(9 − 8x) + C.

Apply a u-substitution with u = cosh x to integrate tanh x, using du = sinh x dx, and obtain ln|cosh x| + C.

Use the substitution u = sqrt x to transform the integral, apply the power rule, and substitute back to obtain 8 sinh(sqrt x) + C.

Rewrite the integrand using the hyperbolic cosine definition, then integrate exponential terms and apply the bounds from 0 to ln 2 to obtain the definite integral.

Evaluate the definite integral of tanh x from 0 to ln 3 by substituting u = cosh x, changing limits to 1 and 5/3, yielding ln(5/3).

Evaluate the limit as x approaches zero of sinh(x)/x using L'Hôpital's rule, showing cosh(0)=1, and confirm the result matches the standard limit: sinh x / x = 1.

Compute the tangent line to f(x) = x cosh x at (1,1) using implicit differentiation, yielding y = cosh(1) x - cosh(1) + 1.

Find the tangent line to the function at (3,0) by computing f'(3) as -6 and using the point-slope form to obtain y = -6x + 18.

Compute derivative via chain rule to find slope of the tangent at (0,5); with m = -10, apply the point slope formula to get y = -10x + 5.

Prove that the derivative of cosh x equals sinh x by expressing cosh x as (e^x+e^{-x})/2 and applying linearity and the chain rule.

Graph the hyperbolic cosine of x by visualizing it as the average of e^x and e^{-x}, then connect plotted points to form the curve.

Apply the second derivative test to locate and classify relative extrema of a function using the product and chain rules, with hyperbolic functions.

Derive the inverse of the hyperbolic sine by solving a quadratic in e^y, then take the natural log to obtain the inverse as ln(x + sqrt(x^2 + 1)).

Learn how to find the area between two graphs using calculus by integrating F(x) minus G(x) over [A,B], where G ≤ F.

Compute the area between the curves y = x^2 and y = 2 − x by hand, finding intersections at -2 and 1 and integrating from -2 to 1.

determine where the square root and the linear function intersect, then compute the area between graphs from 0 to 4 by integrating the top minus bottom, yielding 4/3.

Compute the area between the graphs from zero to one using the definite integral of 3 x e^{-x^2}, top minus bottom, with a u-substitution and updated limits.

Find the area between f(x)=1/(6(1+x^2)) and g(x)=(1/12)x^2 by intersections at x=±1. Use symmetry to compute twice the integral from 0 to 1 of (f−g), yielding pi/12−1/18.

Find the area between the exponential function seven to the x and a straight line from zero to one by a definite integral of the top minus bottom.

Calculate the area between the cube root of x minus seven and the line x minus seven by using intersection points at x = 6, 7, and 8 and symmetry.

Find area between the curves by integrating with respect to y. Intersections at (1,1) and (-2,4) yield a two-part integral from y=0 to 1 and 1 to 4.