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  • need solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise 18.9 question 43

need solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise 18.9 question 43

Answers (1)

Answer:  -\frac{1}{2 x}-\frac{1}{4} \sin \left(\frac{2}{x}\right)+c

Hint: Use substitution method to solve this integral.

Given:   \int \frac{1}{x^{2}} \cos ^{2}\left(\frac{1}{x}\right) d x

Solution:

        \begin{aligned} &\text { Let } I=\int \frac{1}{x^{2}} \cos ^{2}\left(\frac{1}{x}\right) d x \\ &\text { Put } \frac{1}{x}=t \Rightarrow-\frac{1}{x^{2}} d x=d t \\ &\Rightarrow d x=-x^{2} \text { dt then } \end{aligned}

        I=\int \frac{1}{x^{2}} \cdot \cos ^{2} t \cdot\left(-x^{2}\right) d t=-\int \cos ^{2} t \; d t

        =-\int\left\{\frac{1+\cos 2 t}{2}\right\} d t                                                    \left[\begin{array}{l} \because 2 \cos ^{2} A-1=\cos 2 A \\ \Rightarrow 2 \cos ^{2} A=1+\cos 2 A \\ \Rightarrow \cos ^{2} A=\frac{1+\cos 2 A}{2} \end{array}\right]

        \begin{aligned} &=-\int\left\{\frac{1}{2}+\frac{\cos 2 t}{2}\right\} d t=-\int \frac{1}{2} d t-\frac{1}{2} \int \cos 2 t d t \\ &=-\frac{1}{2} \int 1 . d t-\frac{1}{2} \int \cos 2 t d t=-\frac{1}{2} \int t^{0} d t-\frac{1}{2} \int \cos 2 t d t \end{aligned}

        =-\frac{1}{2} t-\frac{1}{4} \sin 2\; t                                                                \left[\begin{array}{c} \because \int x^{n} d x=\frac{x^{n+1}}{n+1}+c \\ \int \cos a\; x \; d x=\frac{\sin a x}{a}+c \end{array}\right]

        =-\frac{1}{2 x}-\frac{1}{4} \sin \left(\frac{2}{x}\right)+c

        

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