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

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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


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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