Evaluate ∫ e^xsin x dx.

Indefinite Integrals · Question 88 of 90

Question 88

HardOriginal question

Evaluate $\int e^{x}\sin x\,dx$ .

\frac{e^{x}(\sin x - \cos x)}{2} + C

Correct answerYour answer

\frac{e^{x}(\sin x + \cos x)}{2} + C

Correct answerYour answer

e^{x}(\sin x - \cos x) + C

Correct answerYour answer

-e^{x}\cos x + C

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

By parts twice: $I = e^{x}\sin x - \int e^{x}\cos x\,dx = e^{x}\sin x - e^{x}\cos x - I$ .
So $2I = e^{x}(\sin x - \cos x)$ and $I = \frac{e^{x}(\sin x - \cos x)}{2} + C$ .

Worth noting

$\frac{e^{x}(\sin x + \cos x)}{2}$ differentiates to $e^{x}\cos x$ , the other integral of the pair.

Ten questions from Indefinite Integrals, at your own pace.

Ten questions from Indefinite Integrals, at your own pace.