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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 34

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 34

Answers (1)

Answer:e^{ax} f(x)+c

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

Given: \int e^{ax}\left [ af\left ( x \right )+{f}'\left ( x \right ) \right ]dx

Solution:

a\int e^{ax} f \left( x \right )dx+\int e^{ax}{f}'\left ( x \right ) dx

Now use integration by parts

                \begin{aligned} &{\left[\int u v d x=u \int v d x-\int\left[\frac{d}{d x} u \int v d x\right] d x+c\right]} \\ &=a\left[f(x) \int e^{a x} d x-\int \frac{d}{d x} f(x) \int e^{a x} d x\right] d x+\int e^{a x} f^{1}(x) d x \\ &=a f(x) \frac{e^{a x}}{a}-a \int f^{1}(x) \frac{e^{a x}}{a} d x+a \int e^{a x} f^{1}(x) d x \\ &=f(x) \cdot e^{a x}-\int f^{1}(x) e^{a x} d x+\int f^{1}(x) e^{a x} d x \\ &=f(x) \cdot e^{a x}+c \end{aligned}

 

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