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