WBJEE · Maths · Definite Integration
If \(f(x)=\int_0^{\sin ^2 x} \sin ^{-1} \sqrt{t}\) \(dt\) and \(g(x)=\int_0^{\cos ^2 x} \cos ^{-1} \sqrt{t}\) \(dt\), then the value of \(f(x)+g(x)\) is
- A \(\pi\)
- B \(\frac{\pi}{4}\)
- C \(\frac{\pi}{2}\)
- D \(\sin ^2 x+\sin x+x\)
Answer & Solution
Correct Answer
(B) \(\frac{\pi}{4}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} \text f^{\prime}(x)+g^{\prime}(x)=0 \\ & \Rightarrow f(x)+g(x)=\text { constant }(\text { say } c) \\ & \text { putting } x=\frac{\pi}{4} \\ \therefore & c=\int_0^{1 / 2}\left(\sin ^{-1} \sqrt{t}+\cos ^{-1} \sqrt{t}\right) d t=\frac{\pi}{4}\end{aligned}\)
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