Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 32

Answer: $\frac{e^{ax}}{a^{2}+b^{2}}\left [ b\cos bx+a\cos bx \right ]+c$

Hints: You must know about the integral rule of trigonometric functions

Given $\int e^{ax}\cos bx dx$

Solution:

$\int e^{ax}\cos bx dx$

Integrating by parts

\begin{aligned} &I=e^{a x} \cdot \frac{\sin b x}{b}-a \int e^{a x} \frac{\sin b x}{b} d x \\ &=\frac{1}{b} e^{a x} \sin b x-\frac{a}{b} \int e^{a x} \sin b x d x \end{aligned}

Again using integration by parts

\begin{aligned} &\frac{1}{b} e^{a x} \sin b x-\frac{a}{b}\left[-e^{a x} \frac{\cos b x}{b}-a \int e^{a x} \frac{\cos b x}{b} d x\right] \\ &\frac{1}{b} e^{a x} \sin b x-\frac{a}{b^{2}} e^{a x} \cos b x-\frac{a^{2}}{b^{2}} \int e^{a x} \cos b x d x \end{aligned}

On computing,

\begin{aligned} &I=\frac{e^{a x}}{b^{2}}[b \sin b x+a \cos b x]-\frac{a^{2}}{b^{2}} I+c \\ &=\frac{e^{a x}}{a^{2}+b^{2}}[b \sin b x+a \cos b x]+c \end{aligned}