Key to "Solve the integral: ∫e8x(3x)dx="

Elma Wilson

Answered question

2021-11-16

Solve the integral:

\int e^{8 x} (3 x) d x =

Liek1993

Beginner2021-11-17Added 13 answers

\int \sin 3 x e^{8 x} d x = \sin 3 x \int e^{8 x} d x - \int (\frac{d}{d x} (\sin 3 x) \int e^{8 x} d x) d x

Let

I = \sin 3 x (\frac{e^{8 x}}{8} - \int 3 \cos 3 x \frac{e^{8 x}}{8} d x

I = \frac{1}{8} e^{8 x} \sin 3 x - \frac{3}{8} \int e^{8 x} \cos 3 x d x

Similarly

\int e^{8 x} \cos 3 x = \cos 3 x \int e^{8 x} d x - \int (\frac{d}{d x} (\cos 3 x) \int e^{8 x} d x) d x

= \frac{e^{8 x}}{8} \cos 3 x + \frac{3}{8} \int e^{8 x} \sin 3 x d x

= \frac{e^{8 x}}{8} \cos 3 x + \frac{3}{8} I

Substitute this in above integration

I = \frac{e^{8 x}}{8} \sin 3 x - \frac{3}{8} (\frac{e^{8 x}}{8} \cos 3 x + \frac{3}{8} I)

I = \frac{e^{8 x}}{8} \sin 3 x - \frac{3}{64} e^{8 x} \cos 3 x - \frac{9}{64} I

I + \frac{9}{64} I = \frac{e^{8 x}}{8} \sin 3 x - \frac{3}{64} e^{8 x} \cos 3 x

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