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  • explain solution RD Sharma class 12 chapter Indefinite Integrals exercise 18.9 question 24 maths

explain solution RD Sharma class 12 chapter Indefinite Integrals exercise 18.9 question 24 maths

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Answer: -e^{\cos ^{2} x}+c

Hint:Use substitution method to solve this integral.

Given:   \int e^{\cos ^{2} x} \sin 2 x\; d x

Solution:

        \begin{aligned} &\text { Let } I=\int e^{\cos ^{2} x} \sin 2 x d x \\ &\text { Put } \cos ^{2} x=t \Rightarrow 2 \cos x(-\sin x) d x=d t \end{aligned}

        \begin{aligned} &\Rightarrow-(2 \cos x \sin x) d x=d t\\ &\Rightarrow-\sin 2 x\; d x=d t \end{aligned}                [\because \sin 2 x=2 \sin x \; \cos x]

        \begin{aligned} &\text { Then }\\ &I=\int e^{t}(-d t)=-\int e^{t} d t=-e^{t}+c \quad\left[\because \int e^{x} d x=e^{x}+c\right]\\ &=-e^{\cos ^{2} x}+c \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\because t=\cos ^{2} x\right] \end{aligned}

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