What is the integration of “cos(ax + b)”?

General solution for integrals like cos(ax + b)

$$I=\int \cos (a x+b) d x$$

Assume $$ax+b=u$$

$$\begin{aligned}&\Longrightarrow \frac{d}{d x}(a x+b) d x=\frac{d}{d u}(u) d u \\&\Longrightarrow a d x=d u \\&\Longrightarrow dx=\frac{1}{a} du\end{aligned}$$

Substitution of x, gives us,

$$\begin{aligned}&I=\int \frac{\cos (u)}{a} du\end{aligned}$$

$$\begin{aligned}=\frac{\sin (u)}{a}\end{aligned}$$

Substitution of u, gives us,

$$I=\frac{\sin (a x+b)}{a}+C$$

Where C is the arbitrary constant of indefinite integration.

$$\int \cos (2x) dx=\frac{\sin (2x)}{2}+C$$

Autor: Ravi Ranjan Kumar Singh