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Techniques of Integration · Substitution Methods

u-Substitution

∫f(g(x)) g′(x) dx=∫f(u) duwhere u=g(x)\int f(g(x))\, g'(x)\, dx = \int f(u)\, du \quad \text{where } u = g(x)

The reverse of the chain rule. Identify an inner function u = g(x), compute du = g'(x) dx, and substitute.

  • 3 worked examples
  • 16 in Techniques of Integration

Worked examples

Find ∫ 2x cos(x²) dx.
  1. Let u = x², du = 2x dx
  2. ∫ cos(u) du = sin(u) + C
  3. Back-substitute: sin(x²) + C

Answer: sin(x²) + C

2 more worked examplesPremium

Find ∫ eˢⁱⁿˣ cos x dx.

Step-by-step solution locked.

Find ∫ x/(x²+1) dx.

Step-by-step solution locked.

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Common questions

What is the u-Substitution?

The reverse of the chain rule. Identify an inner function u = g(x), compute du = g'(x) dx, and substitute. It is one of the techniques of integration formulas in the CalcRef reference.

How do you use the u-Substitution?

Worked example. Find ∫ 2x cos(x²) dx. Let u = x², du = 2x dx. ∫ cos(u) du = sin(u) + C. Back-substitute: sin(x²) + C. Answer: sin(x²) + C.

Practice

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