Integral of eˣ sin x: Formula, Steps and Examples

The integral of e^x\sin x is \dfrac{1}{2}e^x(\sin x - \cos x) + C. It is the famous "loop" integral: integration by parts never finishes on its own, but after two steps you can solve for the answer with algebra.

Graph of eˣ sin x

How to integrate eˣ sin x step by step

Call the integral I. Use integration by parts with the trig function as u both times.

1. First pass. u = \sin x, dv = e^x\,dx, so du = \cos x\,dx and v = e^x: I = e^x\sin x - \int e^x\cos x\,dx
2. Second pass, on \int e^x\cos x\,dx. Again the trig function is u: u = \cos x, dv = e^x\,dx, so du = -\sin x\,dx, v = e^x: \int e^x\cos x\,dx = e^x\cos x + \int e^x\sin x\,dx = e^x\cos x + I
3. Substitute back. The original integral has returned: I = e^x\sin x - e^x\cos x - I
4. Solve for I. Add I to both sides: 2I = e^x(\sin x - \cos x), so I = \frac{e^x}{2}(\sin x - \cos x) + C

The + C is added at the end, after solving. It works just as well with u = e^x both times. What fails is switching: if you take u = \sin x first and u = e^x second, the second step undoes the first and you get I = I.

Check by differentiating

The general formula

The same two-step loop for e^{ax}\sin bx and e^{ax}\cos bx gives

With a = b = 1 the denominator is 2, matching the result above.

A quicker route with complex numbers. Since e^{ix} = \cos x + i\sin x, e^x\sin x is the imaginary part of e^{(1+i)x}. Integrating gives \frac{e^{(1+i)x}}{1+i} = \frac{(1-i)}{2}e^x(\cos x + i\sin x), whose imaginary part is \frac{e^x}{2}(\sin x - \cos x).

Definite integrals

From 0 to π:

From 0 to π/2: \frac{e^{\pi/2}}{2}(1 - 0) - \frac{1}{2}(0 - 1) = \frac{e^{\pi/2} + 1}{2} \approx 2.905.

Decaying version. \int_0^\infty e^{-x}\sin x\,dx. Use the formula with a = -1, b = 1: the antiderivative is \frac{e^{-x}(-\sin x - \cos x)}{2}, which goes to 0 at infinity. So the value is 0 - \left(-\frac{1}{2}\right) = \frac{1}{2}.

Worked examples

Example 1: eˣ cos x. From step 2 above, \int e^x\cos x\,dx = e^x\cos x + I. Substitute I:

Example 2: different rates. \int e^{2x}\sin 3x\,dx = \frac{e^{2x}(2\sin 3x - 3\cos 3x)}{13} + C, using a^2 + b^2 = 4 + 9 = 13.

Example 3: damped vibration. A spring's displacement is e^{-t}\cos t. Its total signed displacement over [0, \infty) is \int_0^\infty e^{-t}\cos t\,dt = \frac{1}{2}, from the cosine formula with a = -1, b = 1: the antiderivative \frac{e^{-t}(\sin t - \cos t)}{2} goes from -\frac{1}{2} to 0.

Common mistakes

1. Switching the choice of u between the two passes. You end up with I = I, which is true but useless.
2. Forgetting to divide by 2. After step 3 you have 2I, not I.
3. Sign errors in the second pass. \frac{d}{dx}\cos x = -\sin x, so the second integral is added, not subtracted.
4. Expecting parts to terminate. With x\sin x the power of x drops away; with e^x\sin x nothing drops away, so you need the loop. Compare the integral of x sin x.

The repeated parts and LIATE lesson practices this loop, and the integration by parts guide has more examples.