§5.3 Integration by substitution

Introduction

Reversing the Chain Rule: First Steps

Reversing the Chain Rule:

Evaluating Definite Integrals via

Using the substitution for the linear inner function, evaluate $\int \sec^2(4x)\,dx$.

Spotting the function-derivative pair inside, evaluate $\int\frac{x^2}{5x^3+1}\,dx$.

With an exponential paired with its own derivative, evaluate $\int e^x\sin(e^x)\,dx$.

Using the chain rule, the derivative of $q(x) = (2-7x)^4$ is

The general antiderivative of $h(x) = (5x-3)^6$ is

Using a linear substitution on the inner function, evaluate the integral $\int\frac{1}{11x-9}\,dx$.

Evaluate $\int\csc(2x+1)\cot(2x+1)\,dx$.

Evaluate $\int\frac{1}{\sqrt{1-16x^2}}\,dx$.

Treating the integrand as an exponential with base five, evaluate $\int 5^{-x}\,dx$.

Using the substitution that pairs with the inner quartic, evaluate $\int x^3\sin(7x^4+3)\,dx$.

Evaluate $\int\frac{\cos(\sqrt{x})}{\sqrt{x}}\,dx$.

For $\int_0^1 e^{-x}(2e^{-x}+3)^{9}\,dx$, the correct substitution and resulting antiderivative are

Evaluate $\int_1^2\frac{x}{1+4x^2}\,dx$ using $u = 1 + 4x^2$, rounded to three decimals. The exact value is $\frac{1}{8}(\ln 17 - \ln 5)$.

Evaluate $\int_{2/\pi}^{4/\pi}\frac{\cos(\frac{1}{x})}{x^2}\,dx$ using $u = \frac{1}{x}$, given the exact value $1 - \frac{\sqrt{2}}{2}$. Report the decimal, rounded to three places.

A valid antiderivative of $\sin(8 - 3x)$ is $\frac{1}{3}\cos(8 - 3x) + C$.