Skill: integration-techniques

Integration Techniques Practice

Beyond the power rule: u-substitution runs the chain rule backwards, integration by parts trades a hard integral for an easier one, and both unlock area between curves and volumes of revolution.

1 Practice

Type your answer and press Check answer (or Enter). A wrong answer earns a hint; a second miss earns a stronger hint; a third miss walks you through the full solution. Scores and streaks are session-only.

Score 0 · Streak 0 · Problem 0

Score counts first-try correct answers. Streak counts consecutive correct answers.

Problem

Enter integers or decimals. For volume questions enter just k (V = kπ). You may type “pi” for π.

2 What you’ll practice

Every problem is generated fresh from templates across several question types — a problem never repeats until its whole pool is used up.

u-substitution

Spot the inner function; du is hiding in the integrand.

Ex: ∫6x(x²+1)³dx → K = 3/4, M = 4

u-sub with exponential

u = the exponent.

Ex: ∫4xe^(x²+1)dx → K = 2

u-sub definite

Change bounds or back-substitute.

Ex: ∫₀¹2x(x²+1)²dx = 7/3

Integration by parts

LIATE: u = x usually.

Ex: ∫₀¹xeˣdx = 1

Choose u and dv

Which split follows LIATE?

Ex: ∫x·cos x dx → u = x

Area between curves

Crossings first, then top − bottom.

Ex: y=x vs y=x² → 1/6

Volume by disks

V = kπ — radius squared, then read k.

Ex: y=x² on [0,1] → k = 1/5

3 Watch out for these

The mistakes students make most often on these problems.

u chosen so du is missing

Good u-substitutions make du (up to a constant) visible in the integrand — look for the derivative of your u.

Bounds not updated

New variable, new bounds: with u = x²+1, x∈[0,1] becomes u∈[1,2]. Or back-substitute first.

Radius not squared / wrong radius

V = π∫R²: square the y-value, and make sure R is the distance to the axis. Draw the region first.

Review the lesson Open the tool Print a worksheet