Integrate a sum of powers and retain the integration constant.
Notes and key methods
For \(\displaystyle n\ne - 1, \int x^{n}dx= \frac{x^{n+ 1}}{n+ 1}+ C\) on a valid interval. Increase the power, divide by that new power, and add an arbitrary constant. Differentiate the result to check it.
A definite integral is F(b)−F(a), and its arbitrary constant cancels. The \(\displaystyle x^{−1}\) case has a logarithmic primitive and must be handled separately.
Check the conditions
Do not apply the power formula at exponent −1.
Include C for an indefinite integral.
Three worked examples
Example 1 · Reverse differentiation and indefinite integration
The integrand is \(f(x)=3 x^{3} + 2 x\), on real x.
Find the general indefinite integral of the given function.
Evaluate the definite integral from \(0\) to \(2\).
Find the primitive G satisfying \(\displaystyle G(0)=0\).
For the primitive with \(\displaystyle F(0)=0\), find F(1).
Show the reasoning
(a)
Differentiate \(\displaystyle \frac{3 x^{4}}{4} + x^{2}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.
Example 2 · Reverse differentiation and indefinite integration
The integrand is \(f(x)=4 x^{4} + 3 x\), on real x.
Find the general indefinite integral of the given function.
Evaluate the definite integral from \(0\) to \(3\).
Find the primitive G satisfying \(\displaystyle G(0)=0\).
For the primitive with \(\displaystyle F(0)=0\), find F(1).
Show the reasoning
(a)
Differentiate \(\displaystyle \frac{4 x^{5}}{5} + \frac{3 x^{2}}{2}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.