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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 23

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Answer: \frac{a^{x}}{\log a}+\frac{x^{a+1}}{a+1}+c

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

Given: \int\left(e^{x \log _{e} a}+e^{a \log _{e} x}\right) d x \\


\begin{aligned} &\int\left(e^{x \log _{e} a}+e^{a \log _{e} x}\right) d x \\ &=\int e^{x \log _{e} a} d x+e^{a \log _{e} x} d x \\ &=\int e^{\log _{e} a x}+e^{\log_{e}x^{a}} d x \\ &=\int a^{x} d x+\int x^{a} d x \quad\quad\quad\quad\quad\quad\quad\left[e^{\log x}=x\right] \\ &=\frac{a^{x}}{\log a}+\frac{x^{a+1}}{a+1}+c \end{aligned}

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