Substitution Rule for Indefinite Integrals — Question 1

Question 1

Evaluate the integral $$\int \frac{x\,e^{\sqrt{1+x^2}}}{\sqrt{1+x^2}}\,dx.$$

Question 1 - Solution

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

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

Integrate: $$\int e^{u}\,du = e^{u}.$$

Substitute back: $$\boxed{ e^{\sqrt{1+x^2}} + C }$$