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  • Provide Solution For R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.25 Question 33 Maths Textbook Solution.

Provide Solution For R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.25 Question 33 Maths Textbook Solution.

Answers (1)

Answer: =\tan \: x+log\left ( \cos x \right )-\frac{x^{2}}{2}+c

Given: \int x\left ( \frac{\sec 2x-1}{\sec 2x+1} \right )dx

Hint: \int uvdx=u\int vdx-\int \left ( \frac{d}{dx}u\int vdx \right )dx

Solution:

            \begin{aligned} &I=\int x\left(\frac{1-\cos 2 x}{1+\cos 2 x}\right) d x \\ &I=\int x\left(\frac{2 \sin ^{2} x}{2 \cos ^{2} x}\right) d x \\ &I=\int x\left(\sec ^{2} x-1\right) d x \\ &I=\int x \sec ^{2} x-\int x d x \\ &I=x \int \sec ^{2} x-\int\left[\frac{d}{d x} x \int \sec ^{2} x d x\right] d x-\frac{x^{2}}{2}+c \\ &I=x \tan x-\int \tan x d x-\frac{x^{2}}{2}+c \\ &=x \tan x+\log (\cos x)-\frac{x^{2}}{2}+c \end{aligned}

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