Substitution Rule for Indefinite Integrals — Question 5

Question 5 - Solution

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

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

Rewrite the integrand: $$\frac{2u}{1+u} = 2\left(1-\frac{1}{1+u}\right).$$

Integrate: $$\int 2\left(1-\frac{1}{1+u}\right)\,du = 2u-2\ln(1+u).$$

Substitute back: $$\boxed{ 2\sqrt{1+x^2} -2\ln\!\left(1+\sqrt{1+x^2}\right) + C }$$