MHT CET · Maths · Definite Integration
\(\int_{0}^{\frac{\pi}{2}} \frac{\sqrt[7]{\sin x}}{\sqrt[7]{\sin x}+\sqrt[7]{\cos x}} d x=\)
- A \(\frac{\pi}{2}\)
- B \(\frac{\pi}{3}\)
- C \(\frac{\pi}{4}\)
- D \(\frac{\pi}{8}\)
Answer & Solution
Correct Answer
(C) \(\frac{\pi}{4}\)
Step-by-step Solution
Detailed explanation
(D)
\(I=\int_{0}^{\frac{\pi}{2}} \frac{\sqrt[7]{\sin x}}{\sqrt[7]{\sin x}+\sqrt[7]{\cos x}}\)...(1)
\(=\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin \left(\frac{\pi}{2}-x\right)}}{\sqrt{\sin \left(\frac{\pi}{2}-x\right)+\sqrt{\cos \left(\frac{\pi-x}{2}\right)}} \mathrm{dx}}\)
\(I=\int_{0}^{r / 2} \frac{\sqrt[7]{\cos x}}{\sqrt[7]{\cos x}+\sqrt[7]{\sin x}} d x\)...(2)
Equation \((1)-(2)\) gives
\(\begin{aligned}
2 I &=\int_{0}^{\frac{\pi}{2}} 1 d x=[x]_{0}^{\frac{\pi}{2}}=\frac{\pi}{2} \\
\therefore I &=\frac{\pi}{4}
\end{aligned}\)
\(I=\int_{0}^{\frac{\pi}{2}} \frac{\sqrt[7]{\sin x}}{\sqrt[7]{\sin x}+\sqrt[7]{\cos x}}\)...(1)
\(=\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin \left(\frac{\pi}{2}-x\right)}}{\sqrt{\sin \left(\frac{\pi}{2}-x\right)+\sqrt{\cos \left(\frac{\pi-x}{2}\right)}} \mathrm{dx}}\)
\(I=\int_{0}^{r / 2} \frac{\sqrt[7]{\cos x}}{\sqrt[7]{\cos x}+\sqrt[7]{\sin x}} d x\)...(2)
Equation \((1)-(2)\) gives
\(\begin{aligned}
2 I &=\int_{0}^{\frac{\pi}{2}} 1 d x=[x]_{0}^{\frac{\pi}{2}}=\frac{\pi}{2} \\
\therefore I &=\frac{\pi}{4}
\end{aligned}\)
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