Integral Substitution Visualizer

Master u-substitution with 20+ fully worked examples across polynomial, exponential, trigonometric, and logarithmic integrals. Step through each solution at your own pace or reveal all steps at once.

21worked examples

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Example 1 of 21

Polynomial

Evaluate:

Substitution: u = x²+1Differential: du = 2x dx

Step 1: Choose u

Pick the inner expression of the power. The factor 2x outside is its derivative.

Reference Guide

The U-Substitution Method

U-substitution is the integral analogue of the chain rule. When an integrand contains a composite function, letting $u = g(x)$ simplifies the integral into a standard form.

The recipe

Identify a function $u = g(x)$
Compute $du = g'(x)\,dx$
Replace every occurrence of $x$ and $dx$
Integrate in terms of $u$
Substitute back $u = g(x)$

Core formula

\int f(g(x))\,g'(x)\,dx = \int f(u)\,du

Choosing U Wisely

The most important step is picking the right $u$ . A good choice makes $du$ appear (possibly up to a constant factor) elsewhere in the integrand.

Patterns to look for

Inner function of a composite: $\sin(x^2)$ - pick $u=x^2$
Exponent of $e$ : $e^{3x}$ - pick $u=3x$
Denominator when numerator is its derivative: $\frac{2x}{x^2+1}$ - pick $u=x^2+1$
Argument of a trig function: $\cos(x^2)$ - pick $u=x^2$

If your choice forces a constant adjustment (like $x\,dx = \tfrac{1}{2}du$ ), that is fine. Only fail if $x$ cannot be eliminated entirely.

When U-Substitution Works

U-substitution works when you can write the integrand as $f(g(x)) \cdot g'(x)$ for some functions $f$ and $g$ .

Standard forms unlocked

\int u^n\,du = \tfrac{u^{n+1}}{n+1}+C

\int e^u\,du = e^u+C

\int \tfrac{du}{u} = \ln|u|+C

\int \cos(u)\,du = \sin(u)+C

When substitution fails (the remaining $x$ terms cannot be expressed in $u$ ), try integration by parts, trig identities, or partial fractions instead.

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