NIMCET 2020 — Mathematics PYQ

Concept:
- Derivatives of Trigonometric Functions:
$$\frac{d}{dx}\sin x=\cos x$$
$$\frac{d}{dx}\cos x=-\sin x$$
$$\frac{d}{dx}\tan x={\sec }^{2}x$$
$$\frac{d}{dx}\cot x=-{\csc }^{2}x$$
$$\frac{d}{dx}\sec x=\tan x\sec x$$
$$<br>\frac{d}{dx}\csc x=-\cot x\csc x$$
-
$$\int e^x[f(x)+f(x)]dx=e^{x}f(x)+C.$$

Calculation:
$$\int e^x\left(\frac{1+\sin x\cos x}{{\cos }^{2}x}\right)dx$$
$$=\int e^x\left(\frac{1}{{\cos }^{2}x}+\frac{\sin x\cos x}{{\cos }^{2}x}\right)dx$$
$$=\int e^x\left[{\sec }^{2}x+\tan x\right]dx$$
$$=\int e^x\left[f(x)+f^{\prime }(x)\right]dx,\text{where }f(x)=\tan x.$$
$$=e^{x}f(x)+C$$
$$=e^x\tan x+C.$$