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  • provide solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise 18.26 question 22

provide solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise 18.26 question 22

Answers (1)

Answer:
The correct answer is  x \tan (\log x)+c
Hint:

\int e^{x}\left(f(x)+f^{\prime}(x)\right) d x=e^{x} f(x)+c

Given:

\int\left[\tan (\log x)+\sec ^{2}(\log x)\right] d x

Solution:

        I=\int\left[\tan (\log x)+\sec ^{2}(\log x)\right] d x

\begin{aligned} &\text { Put } \log x=t \\ &\Rightarrow x=e^{t} \\ &d x=e^{t} d t \end{aligned}

        I=\int\left(\tan t+\sec ^{2} t\right) e^{t} d t

\begin{aligned} &\text { Here, } f(t)=\tan t \\ &f^{\prime}(t)=\sec ^{2} t \\ &\text { Let } e^{t} \tan t=p \end{aligned}

Differentiate both sides w.r.t ‘t’

        \begin{aligned} &e^{t}\left(\tan t+\sec ^{2} t\right)=\frac{d p}{d t} \\ &e^{t}\left(\tan t+\sec ^{2} t\right) d t=d p \end{aligned}

        \begin{aligned} &I=\int d p \\ &=p+c \\ &=e^{t} \tan t+c \\ &=x \tan (\log x)+c \end{aligned}

So, the correct answer is   x \tan (\log x)+c

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