Integral Exercise 10

We are asked to compute the following integral:

$$ \int 2x(1+x^2)^5 \ dx $$

This is a classic example of an integral of the form fʼ(x) · [f(x)]ⁿ, where $ f(x) = 1 + x^2 $, $ fʼ(x) = 2x $, and $ n = 5 $.

For integrals of this type, we can apply the standard formula:

$$ \int f'(x) \cdot [ f(x) ]^n \ dx = \frac{[f(x)]^{n+1}}{n+1} + c $$

$$ \int 2x(1+x^2)^5 \ dx = \frac{(1+x^2)^{5+1}}{5+1} + c $$

Which simplifies to:

$$ \int 2x(1+x^2)^5 \ dx = \frac{(1+x^2)^6}{6} + c $$

Therefore, the antiderivative is:

$$ F(x) = \frac{(1+x^2)^6}{6} + c $$

And that completes the solution.