For any integer n, the integral (2n + 1)x dx has the value.

JEE MainMathematicsModerateDefinite Integration

For any integer n, the integral
$\int_{0}^{\pi}\mspace{2mu} e^{\sin^{2}x}\cos^{3}$ (2n + 1)x dx has the
value.

Show answer and solution

\because f(x) = e^{\sin^{2}x}\cos^{3}(2n + 1)x

f(\pi - x) = - e^{\sin^{2}x}\cos^{3}(2n + 1)x = - f(x)

Use property

\int_{0}^{2a}\mspace{2mu} f(x)dx = 0

(where f(2a - x) =
- f(x))

Source: Grass-G / M-13 Mathematics Paper 4 Nov.docx

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