- \(\displaystyle \int e^{\sin ^2 (x)} \sin (2x) ~dx = \color{blue} {e^{\sin^2(x)}}\)
Solution: \(u = \sin ^2 (x)\), \(du = 2 \sin (x) \cos (x) ~dx = \sin (2x) ~dx\)
Used 2018
Used 2019
Solution
1. With \(u = \sin^2(x)\), \(du = \sin(2x) \, dx\).
By the chain rule, \(\frac{du}{dx} = 2\sin(x)\cos(x)\), and by the double-angle identity for sine, \(2\sin(x)\cos(x) = \sin(2x)\).
2. Transform and evaluate the integral in terms of \(u\): \[\int e^{\sin^2(x)} \sin(2x) \, dx = \int e^u \, du = e^u + C.\]
By 1, \(\sin(2x)\,dx = du\), so the integral becomes \(\int e^u\,du\); the antiderivative of \(e^u\) is \(e^u + C\).
3. Substitute back \(u = \sin^2(x)\): \[\int e^{\sin^2(x)} \sin(2x) \, dx = e^{\sin^2(x)} + C.\]
Replacing \(u\) by \(\sin^2(x)\) in the antiderivative \(e^u + C\) from 2 gives \(e^{\sin^2(x)} + C\).