Substitution

The method

Definite integrals

When algebra must prepare the way

Connections and checks

Which condition must hold for the substitution u = g(x) to convert an integral directly into a basic table entry?

For the integral of x^2 sqrt(x^3 + 1) dx, which substitution choice converts it into a pure power rule integral?

In the definite integral from 0 to 1 of 2x (x^2 + 1)^3 dx with u = x^2 + 1, what are the new limits of integration?

The integral of tan x dx is evaluated by rewriting tan x as sin x over cos x. Which substitution completes it?

After finding an indefinite integral by substitution, what is the most reliable check of the answer?

For which integrand would the substitution u = x^3 + 1 NOT directly apply?

With u = x^2 + 1, the integral from 0 to 1 of 2x (x^2 + 1) dx becomes the integral of u du from 1 to 2. What is its value? Answer with a fraction like 3/2.

The integral of x e^(x^2) dx uses u = x^2 with du = 2x dx. What constant factor must sit outside the transformed integral? Answer with a fraction like 1/2.

The integral of 1/(x ln x) dx with u = ln x becomes the integral of du/u. What is the result? Answer like ln|u| + C.

The integral of tan x dx equals the negative logarithm of which function, plus C? Answer with the function like cos x.