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Range of Trigonometric Expressions

sin 2 x d x

Question

$\int e^{x} \sin^{2} x d x$

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Solution

$Let I = \int e^{x} \sin^{2} x d x = \int e^{x} (\frac{1 - \cos 2 x}{2}) d x = \frac{1}{2} \int e^{x} d x - \frac{1}{2} \int e^{x} \cos 2 x d x = \frac{e^{x}}{2} - \frac{1}{2} \int e^{x} \cos (2 x) d x . . . . . (1) Let I_{1} = \int e^{x} \cos (2 x) d x Considering \cos (2 x) as first function and e^{x} as second function I_{1} = \cos (2 x) e^{x} - \int - 2 \sin (2 x) e^{x} d x \Rightarrow I_{1} = \cos (2 x) e^{x} + 2 \int \sin (2 x) e^{x} d x \Rightarrow I_{1} = \cos (2 x) e^{x} + 2 [\sin (2 x) e^{x} - \int 2 \cos (2 x) e^{x} d x] \Rightarrow I_{1} = \cos (2 x) e^{x} + 2 \sin (2 x) e^{x} - 4 I_{1} \Rightarrow 5 I_{1} = e^{x} (\cos 2 x + 2 \sin 2 x) \Rightarrow I_{1} = \frac{e^{x}}{5} (\cos 2 x + 2 \sin 2 x) + C . . . . . (2) From (1) & (2) I = \frac{e^{x}}{2} - \frac{e^{x}}{10} (\cos 2 x + 2 \sin 2 x) + C$

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