Substitution Rule for Definite Integrals — Question 8

Question 8

Evaluate the definite integral $$\int_{0}^{1} \frac{2x}{\left(1+x^2\right)^2}\,dx.$$

Question 8 - Solution

The integrand suggests a substitution involving $1+x^2$.

Let $$u=1+x^2.$$ Then $$du=2x\,dx.$$

Change the limits of integration. When $x=0$, $u=1$. When $x=1$, $u=2$.

Substitute into the integral: $$\int_{0}^{1} \frac{2x}{(1+x^2)^2}\,dx = \int_{1}^{2} u^{-2}\,du.$$

Integrate: $$\int u^{-2}\,du = -\,u^{-1}.$$

Apply the limits: $$-\,u^{-1}\Big|_{1}^{2} = -\frac{1}{2}-(-1) = \frac{1}{2}.$$

$$\boxed{\frac{1}{2}}$$