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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 31

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 31

Answers (1)

Answer: \frac{e^{ax}}{a^{2}+b^{2}}\left ( a\sin bx-b\cos bx \right )+c

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

Given: \int e^{ax}\sin bxdx

Solution:

I=\int e^{ax}\sin bxdx and using integration by parts,

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

\begin{aligned} &I\left(1+\frac{b^{2}}{a^{2}}\right)=\frac{1}{a} \sin b x \cdot e^{a x}-\frac{b}{a^{2}} \cos b x \cdot e^{a x} \\ &\Rightarrow \int e^{a x} \cdot \sin b x d x=\frac{e^{a x}}{a^{2}+b^{2}}(a \sin b x-b \cos b x)+c \end{aligned}

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