AP EAMCET · Maths · Definite Integration
If \(f(x)=\sin \left(\tan ^{-1} x\right)\), then \(\int_0^1 x f^{\prime \prime}(x) d x=\)
- A \(1-\frac{3}{2 \sqrt{2}}\)
- B \(-\frac{1}{2 \sqrt{2}}\)
- C \(\frac{1}{\sqrt{2}}\)
- D \(-\sqrt{2}\)
Answer & Solution
Correct Answer
(B) \(-\frac{1}{2 \sqrt{2}}\)
Step-by-step Solution
Detailed explanation
\begin{aligned} f(x) & =\sin \left(\tan ^{-1} x\right) \\ f^{\prime}(x) & =\frac{\cos \left(\tan ^{-1} x\right)}{x^2+1} \\ I & =\int_0^1 x \cdot f^{\prime \prime}(x) d x={ }_0^1\left[x \cdot f^{\prime}(x)\right]-\int_0^1 f^{\prime}(x) d x \\ & =f^{\prime}(1)-{…
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