Substitution

SyNtHeSiS

Evaluate the indefinite integral:

∫ (1 + x) / (1 + x^2) dx

Random Variable

$$\displaystyle = \int \frac{1}{1+x^{2}} \ dx + \int \frac{x}{1+x^{2}} \ dx$$

HallsofIvy

SyNtHeSiS

$$\displaystyle = \int \frac{1}{1+x^{2}} \ dx + \int \frac{x}{1+x^{2}} \ dx$$
I managed to integrate the right term of that sum and got 1/2ln|1 + x^2| + C, but I am struggling to integrate the left part. I have made 3 substitutions, and have almost got it, but there are always 2 variables in the integral.

Prove It

MHF Helper
I managed to integrate the right term of that sum and got 1/2ln|1 + x^2| + C, but I am struggling to integrate the left part. I have made 3 substitutions, and have almost got it, but there are always 2 variables in the integral.
You use a trigonometric substitution $$\displaystyle x = \tan{\theta}$$, so that $$\displaystyle dx = \sec^2{\theta}\,d\theta$$.

Note too that $$\displaystyle \theta = \arctan{x}$$.

Therefore

$$\displaystyle \int{\frac{1}{1 + x^2}\,dx} = \int{\frac{1}{1 + \tan^2{\theta}}\,\sec^2{\theta}\,d\theta}$$

$$\displaystyle = \int{\frac{\sec^2{\theta}}{\sec^2{\theta}}\,d\theta}$$

$$\displaystyle = \int{1\,d\theta}$$

$$\displaystyle = \theta + C$$

$$\displaystyle = \arctan{x} + C$$.