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

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

Answers (1)

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} 

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