Integrals · Definition

Indefinite integral

calculusintegrals

Before you start you need to know

Integrating is the inverse of differentiating: the indefinite integral collects all antiderivatives of a function.

Definition

The indefinite integral of \(f\) is the family of all its antiderivatives, written with a constant of integration \(C\).

\[\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C\quad(n\neq-1)\]

Linearity: \(\int(af+bg)\,dx=a\int f\,dx+b\int g\,dx\); constants come out of the integral, which is why we integrate term by term.

Guided example: \(\int(3x^{2}+\frac{1}{x})\,dx=x^{3}+\ln|x|+C\), because \(\int 3x^{2}\,dx=x^{3}\) and \(\int\frac{1}{x}\,dx=\ln|x|\).

Typical error: forgetting the constant \(C\), or writing \(\int x^{n}\,dx=\frac{x^{n+1}}{n+1}\) without dividing by \(n+1\). To check, differentiate the result.

Think about it

Compute \(\int(3x^2+1)\,dx\).

\(x^3+x+C\).

Exercises 11

Compute the following integrals

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

Problems

  1. What is \(\int \cos x\,dx\)?
  2. Compute \(\int \frac{x^{2}+1}{x}\,dx\) for \(x>0\).

References

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