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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 26

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Answer:  \frac{\log\left | 3+a^{x} \right |}{\ln \left ( a \right )}+c

Hints: You must know about the integral rules of x

Given: \int \frac{a^{x}}{3+a^{x}}dx


I=\int \frac{a^{x}}{3+a^{x}}dx

Lett= 3+ a^{x}.         and differentiate both sides

d t=0+a^{x} \ln (a) d x \quad\quad \quad \quad \quad \quad \quad \quad{\left[\frac{d}{d x} a^{x}=a^{x} \ln |a|\right]} \\

I=\int \frac{d t}{\ln (a) t} \quad \quad \quad \quad \quad \quad \quad \quad {\left[\int \frac{1}{x} d x=\log x+c\right]}

 \begin{aligned} &=\frac{1}{\ln a} \int \frac{d t}{t} \\ &=\frac{1}{\ln a} \log |t|+c \\ &=\frac{\log \left|3+a^{x}\right|}{\ln (a)}+c \end{aligned}

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