Integrals · Definition

Integration by substitution

calculusintegrals

Before you start you need to know

It is useful when the integrand is a composite function times the derivative of its inner part: the change \(u=g(x)\) makes it a simpler integral.

Definition

Technique that simplifies an integral through the change of variable \(u=g(x)\), \(du=g^{\prime}(x)\,dx\).

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

Substitution is the chain rule in reverse. If inside the integrand you see a function and, multiplying, its derivative, change to a new variable: with \(u=g(x)\) and \(du=g'(x)\,dx\) the integral becomes simpler.

Example: \(\int 2x\cos(x^2)\,dx\). With \(u=x^2\), \(du=2x\,dx\) and you get \(\int\cos u\,du=\sin(x^2)+C\).

Typical error: forgetting to replace \(dx\) by \(du\) or to change the limits in a definite integral; the result is then wrong.

Think about it

Compute \(\int 2x\,e^{x^2}\,dx\).

With \(u=x^2\): \(e^{u}+C=e^{x^2}+C\).

Exercises 10

Compute the following integrals

  1. \(\displaystyle \int 2 x \cos {x^{2}}\,dx\)
  2. \(\displaystyle \int x e^{x^{2}}\,dx\)
  3. \(\displaystyle \int \frac{3 x^{2}}{x^{3} + 1}\,dx\)
  4. \(\displaystyle \int \sin {x} \cos^{2} {x}\,dx\)
  5. \(\displaystyle \int \left(2 x + 1\right) \left(x^{2} + x\right)^{4}\,dx\)
  6. \(\displaystyle \int e^{3 x}\,dx\)
  7. \(\displaystyle \int_{0}^{1} \frac{x}{1+x^{2}}\,dx\)
  8. \(\displaystyle \int \frac{\ln x}{x}\,dx\)

Problems

  1. When is it convenient to use integration by substitution?
  2. If \(u=g(x)\), what is the differential \(du\)?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.