MHT CET · Maths · Indefinite Integration
\(\int e^{\cos ^{-1} x}\left[\frac{x-\sqrt{1-x^{2}}}{\sqrt{1-x^{2}}}\right] d x=\)
- A \(-e^{\sin ^{-1} x}+c\)
- B \(-x e^{\cos ^{-1} x}+c\)
- C \(-x e^{\sin ^{-1} x}+c\)
- D \(-e^{\cos ^{-1} x}+c\)
Answer & Solution
Correct Answer
(B) \(-x e^{\cos ^{-1} x}+c\)
Step-by-step Solution
Detailed explanation
Put \(\cos ^{-1} x=t \Rightarrow \frac{-1}{\sqrt{1-x^{2}}} d x=d t \Rightarrow \frac{1}{\sqrt{1-x^{2}}} d x=-d t\) and \(x=\cos t\)
\(\begin{aligned} I &=-\int e^{t}[\cos t-\sin t] d t=-e^{t} \cos t+c \\ &=-e^{\cos ^{-1} x} \cos \left(\cos ^{-1} x\right)+c=-x e^{\cos ^{-1} x}+c \end{aligned}\)
\(\begin{aligned} I &=-\int e^{t}[\cos t-\sin t] d t=-e^{t} \cos t+c \\ &=-e^{\cos ^{-1} x} \cos \left(\cos ^{-1} x\right)+c=-x e^{\cos ^{-1} x}+c \end{aligned}\)
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