JEE Mains · Maths · STD 12 - 7.1 indefinite integral
If \(I(x)=\int e^{\sin ^2 x}(\cos x \sin 2 x-\sin x) d x \quad\) and \(I(0)=1\), then \(I\left(\frac{\pi}{3}\right)\) is equal to
- A \(-\frac{1}{2} e^{\frac{3}{4}}\)
- B \(e^{\frac{3}{4}}\)
- C \(\frac{1}{2} e^{\frac{3}{4}}\)
- D \(-e^{\frac{3}{4}}\)
Answer & Solution
Correct Answer
(C) \(\frac{1}{2} e^{\frac{3}{4}}\)
Step-by-step Solution
Detailed explanation
\(I(x)=\int \frac{e^{\sin x} \cdot \sin 2 x}{I I} \cdot \frac{\cos x}{I} d x-\int e^{\sin ^2 x} \cdot \sin x d x\) \(\Rightarrow I(x)=e^{\sin ^2 x}-\int(-\sin x) \cdot e^{\sin ^2 x} d x-\int e^{\sin ^2 x} \cdot \sin x d x\) \(\Rightarrow I(x)=e^{\sin ^2 x} \cdot \cos x+c\) Put…
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