AP EAMCET · Maths · Definite Integration
3. \(\int_0^{\frac{1}{2}} \frac{x \sin ^{-1} x}{\sqrt{1-x^2}} d x=\)
- A \(\left(\frac{1}{2}+\frac{\sqrt{3}}{12} \pi\right)\)
- B \(\left(\frac{1}{2}-\frac{\sqrt{3}}{12} \pi\right)\)
- C \(\left(-\frac{1}{2}+\frac{\sqrt{3}}{12} \pi\right)\)
- D \(\left(-\frac{1}{2}-\frac{\sqrt{3}}{12} \pi\right)\)
Answer & Solution
Correct Answer
(B) \(\left(\frac{1}{2}-\frac{\sqrt{3}}{12} \pi\right)\)
Step-by-step Solution
Detailed explanation
\(\int_0^{1 / 2} \frac{x \sin ^{-1} x}{\sqrt{1-x^2}} d x=\int_0^{1 / 2} x \sin ^{-1} x \cdot \frac{1}{\sqrt{1-x^2}} d x\) ...(i) Let \(\sin ^{-1} x=t \Rightarrow \sin t=x\) Substituting above values in eqn. (i), we get :…
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