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  • Explain solution RD Sharma class 12 chapter 19 Definite Integrals exercise Multiple choice question 12 maths

Explain solution RD Sharma class 12 chapter 19 Definite Integrals exercise Multiple choice question 12 maths

Answers (1)

Answer:

\frac{\pi }{12}

Given:

\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1+\sqrt{\cot x}} d x

Hint:

Using \int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x


Explanation:  

Let

\begin{aligned} &I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1+\sqrt{\cot x}} d x \\\\ &I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1+\sqrt{\frac{\cos x}{\sin x}}} d x, \quad\left[\therefore \cot x=\frac{\cos x}{\sin x}\right] \end{aligned}

    =\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x \ldots(i)

We have a property that

\int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x

Using this, we get

=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\cos \left(\frac{\pi}{3}+\frac{\pi}{6}-x\right)}}{\sqrt{\cos \left(\frac{\pi}{3}+\frac{\pi}{6}-x\right)}+\sqrt{\sin \left(\frac{\pi}{3}+\frac{\pi}{6}-x\right)}} d x

=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} d x,\left[\therefore \sin \left(\frac{\pi}{2}-x\right)=\cos x, \quad \cos \left(\frac{\pi}{2}-x\right)=\sin x\right]

I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\cos x}}{\sqrt{\cos x}+\sqrt{\sin x}} d x \ldots(i i)

Adding (i) and (ii)

\begin{aligned} &2 I=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\sin x}+\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x \\\\ &=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} 1 d x \end{aligned}

\begin{aligned} &I=\frac{1}{2}[x]_{\frac{\pi}{6}}^{\frac{\pi}{3}} \\\\ &I=\frac{1}{2}\left[\frac{\pi}{3}-\frac{\pi}{6}\right] \end{aligned}

\begin{aligned} &I=\frac{1}{2}\left[\frac{2 \pi-2 \pi}{6}\right] \\\\ &I=\frac{1}{2} \times \frac{\pi}{6} \\\\ &=\frac{\pi}{12} \end{aligned}

 

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infoexpert26

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