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Recognizing patterns

Substitution

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

u=g(x),du=g′(x) dxu=g(x),\quad du=g^{\prime}(x)\,dx

Look for a composite function accompanied by its derivative. Define u, compute du and transform the whole integral into u: do not mix x and u. For an indefinite integral, return to x at the end. For a definite integral evaluated in u, transform both bounds using g.

Choose a case

Trigonometric substitution: three radical formsa2−x2,a2+x2,x2−a2\sqrt{a^2-x^2},\quad\sqrt{a^2+x^2},\quad\sqrt{x^2-a^2}View explanation and example

Common mistake

Changing the function without changing dx, or using x bounds in an integral with respect to u.

DEVELOPED GUIDE · 7 min

Choose u and transform bounds without mixing the two variables.

When to choose this method

  • A composite function is accompanied by its derivative, possibly with a different constant factor.
  • For definite integrals, evaluate in u with new bounds or return to x with original bounds.

Before calculating

  1. Define u and identify the product replacing dx.
  2. Transform both endpoints and write an integral entirely in u.

A complete example, step by step

∫012xcos⁡(x2+1) dx\int_0^1 2x\cos(x^2+1)\,dx

  1. The cosine argument has derivative 2x, matching the outer factor.

    u=x2+1,du=2x dxu=x^2+1,\qquad du=2x\,dx

  2. Convert the lower and upper endpoints separately.

    x=0⇒u=1,x=1⇒u=2x=0\Rightarrow u=1,\qquad x=1\Rightarrow u=2

  3. After substitution, no x remains in the integral.

    I=∫12cos⁡u duI=\int_1^2\cos u\,du

  4. Integrate cosine and evaluate upper minus lower.

    I=[sin⁡u]12=sin⁡2−sin⁡1I=[\sin u]_1^2=\sin2-\sin1

Check the result and domain

On [0,1], u increases from 1 to 2. Differentiate sin(x²+1) to check the primitive before evaluating.

ddxsin⁡(x2+1)=2xcos⁡(x2+1)\frac{d}{dx}\sin(x^2+1)=2x\cos(x^2+1)
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Now try it yourself

Solve on paper before opening the hint or answer. These are self-assessment activities; to check a typed answer, open Learn.

EXERCISE 1∫3x2ex3 dx\int 3x^2e^{x^3}\,dx
Show a hint

Try u=x³.

Check my resultex3+Ce^{x^3}+C
EXERCISE 2∫2x1+x2 dx\int\frac{2x}{1+x^2}\,dx
Show a hint

The numerator is the denominator derivative.

Check my resultln⁡(1+x2)+C\ln(1+x^2)+C
EXERCISE 3∫xcos⁡(x2) dx\int x\cos(x^2)\,dx
Show a hint

du=2x dx: a factor 1/2 appears.

Check my result12sin⁡(x2)+C\frac12\sin(x^2)+C
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Connect with other methods

Direct primitives and the chain rule →Integration by parts →Trigonometric substitution: three radical forms →

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