§5.3 Integration by substitution

Introduction

Reversing the Chain Rule: First Steps

Reversing the Chain Rule:

Evaluating Definite Integrals via

In Section 2.5, we learned the Chain Rule and how it can be applied to find the derivative of a composite function. In particular, if $u$ is a differentiable function of $x$, and $f$ is a differentiable function of $u(x)$, then

.

In words, we say that the derivative of a composite function $c(x) = f(u(x))$, where $f$ is considered the “outer” function and $u$ the “inner” function, is “the derivative of the outer function, evaluated at the inner function, times the derivative of the inner function.”

For each of the following functions, use the Chain Rule to find the function's derivative. Be sure to label each derivative by name (e.g., the derivative of $g(x)$ should be labeled $g'(x)$).

-$g(x) = e^{3x}$-$h(x) = \sin(5x+1)$-$p(x) = \arctan(2x)$-$q(x) = (2-7x)^4$-$r(x) = 3^{4-11x}$

Determine the general antiderivative of

. Check the result by differentiating.

$\displaystyle \int \sin(8-3x) \, dx$

$\displaystyle \int \sec^2 (4x) \, dx$

$\displaystyle \int \frac{1}{11x - 9} \, dx$

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

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

$\displaystyle \int 5^{-x}\, dx$

Evaluate the indefinite integral

and check the result by differentiating.

$\displaystyle \int \frac{x^2}{5x^3+1} \, dx$

$\displaystyle \int e^x \sin(e^x) \, dx$

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

$\displaystyle \int_1^2 \frac{x}{1 + 4x^2} \, dx$

$\displaystyle \int_0^1 e^{-x} (2e^{-x}+3)^{9} \, dx$

$\displaystyle \int_{2/\pi}^{4/\pi} \frac{\cos\left(\frac{1}{x}\right)}{x^{2}} \,dx$