Solved: "Evaluate the integral. (Use C for the constant..."

Ann Tice

Answered question

2021-11-15

Evaluate the integral. (Use C for the constant of integration.)

\int e^{4 θ} \sin (5 θ) d θ

Fearen

Beginner2021-11-16Added 15 answers

We have to find the value of integral

\int^{4 θ} \sin (5 θ) d θ

To find the value we use the formula ILATE which states

\int u v = u \int v d x - \int [\frac{d u}{d x}, \int v] d x

Using ILATE we get

u = \sin 5 θ

and

v = e^{4 θ}

substituting the values in the above formula we get

\sin 5 θ e^{4 θ} d θ = \sin 5 θ \int e^{4 θ} d x

\int \sin 5 θ e^{4 θ} d θ = \sin 5 θ \frac{e^{4 θ}}{4} - (\frac{5 e^{4 θ} \cos (5 θ)}{16} + \frac{25}{16} \int \sin 5 θ e^{4 θ} d θ)

\int \sin 5 θ e^{4 θ} d θ + \frac{25}{16} \int \sin 5 θ e^{4 θ} d θ

\int \sin 5 θ e^{4 θ} d θ = \frac{4 \sin 5 θ e^{4 θ} \cos (5 θ)}{41}

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