Substitution Rule for Indefinite Integrals — Question 6

Question 6 - Solution

Use substitution. Let $$u=\frac{1}{1+x^2}.$$ Then $$du=-\frac{2x}{(1+x^2)^2}\,dx \quad\Rightarrow\quad -\frac{1}{2}\,du=\frac{x}{(1+x^2)^2}\,dx.$$

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

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

Substitute back: $$\boxed{ -\frac{1}{2}e^{1/(1+x^2)}+C }$$