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  • Explain Solution R.D. Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 65 Maths Textbook Solution.

Explain Solution R.D. Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 65  Maths Textbook Solution.

Answers (1)

Answer:

I=\frac{1}{\sqrt{3}} \log \left|\frac{\tan \left(\frac{x}{2}\right)-2-\sqrt{3}}{\tan \left(\frac{x}{2}\right)-2+\sqrt{3}}\right|+c

Hint:

To solve the above equation we will write sin x as \frac{2 \tan \left(\frac{x}{2}\right)}{1+\tan ^{2}\left(\frac{x}{2}\right)}

Given:

\int \frac{1}{1-2 \sin x} d x

Solution: 

I=\int \frac{1}{1-2 \sin x} d x

    I=\int \frac{1}{1-\frac{4 \tan \left(\frac{x}{2}\right)}{1+\tan ^{2}\left(\frac{x}{2}\right)}} d x

I=\int \frac{1+\tan ^{2}\left(\frac{x}{2}\right)}{1+\tan ^{2}\left(\frac{x}{2}\right)-4 \tan \left(\frac{x}{2}\right)} d x                                        \left[\because \sin x=\frac{2 \tan \left(\frac{x}{2}\right)}{1+\tan ^{2}\left(\frac{x}{2}\right)}\right.

I=\int \frac{\sec ^{2}\left(\frac{x}{2}\right)}{1+\tan ^{2}\left(\frac{x}{2}\right)-4 \tan \left(\frac{x}{2}\right)} d x

I=\int \frac{2}{2} \frac{\sec ^{2}\left(\frac{x}{2}\right)}{1+\tan ^{2}\left(\frac{x}{2}\right)-4 \tan \left(\frac{x}{2}\right)} d x

I=2 \int \frac{d t}{1+t^{2}-4 t}                                \left[\because t=\tan \left(\frac{x}{2}\right), \frac{d t}{d x}=\frac{1}{2} \sec ^{2}\left(\frac{x}{2}\right), d t=\frac{1}{2} \sec ^{2}\left(\frac{x}{2}\right) d x\right]

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

I=2 \int \frac{d t}{(t-2)^{2}-3}

I=2 \int \frac{d u}{u^{2}-(\sqrt{3})^{2}}                            [\because t-2=u, d t=d u]

\left[\int \frac{d x}{x^{2}-a^{2}}=\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+c\right.

I=2 \times \frac{1}{2 \sqrt{3}} \log \left|\frac{u-\sqrt{3}}{u+\sqrt{3}}\right|+c

I=\frac{1}{\sqrt{3}} \log \left|\frac{t-2-\sqrt{3}}{t-2+\sqrt{3}}\right|+c

I=\frac{1}{\sqrt{3}} \log \left|\frac{\tan \left(\frac{x}{2}\right)-2-\sqrt{3}}{\tan \left(\frac{x}{2}\right)-2+\sqrt{3}}\right|+c

 

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