Cyclic integration by parts

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

I=A-kI

For certain exponential and sine or cosine products, the original integral reappears. Name it I, retain its signs and solve algebraically.

Do not repeat indefinitely: solve when I reappears.

DEVELOPED GUIDE · 8 min

Close the integration-by-parts cycle and isolate the recurring integral.

When to choose this method

An exponential times sine or cosine returns to its original form after two applications.
Keep the same symbol I throughout and add C once at the end.

I=\int e^x\sin(2x)\,dx

Choose u=sin(2x) and dv=eˣ dx; the derivative of u includes a factor 2.
$du=2\cos(2x)\,dx,\quad v=e^x$
Apply the formula and call the remaining integral J.
$I=e^x\sin(2x)-2J,\qquad J=\int e^x\cos(2x)\,dx$
Integrate J by parts again. The cosine derivative has a minus sign.
$J=e^x\cos(2x)+2I$
Substitute J into the first equation. The −2 multiplies every term.
$I=e^x\sin(2x)-2e^x\cos(2x)-4I$
Add 4I to both sides and divide by 5.
$I=\frac{e^x}{5}\bigl(\sin(2x)-2\cos(2x)\bigr)+C$

The product rule produces cancelling terms and recovers eˣ sin(2x). The formula is valid for all real x.

F'(x)=\frac{e^x}{5}\,[\sin(2x)-2\cos(2x)+2\cos(2x)+4\sin(2x)]=e^x\sin(2x)

Solve on paper before opening the hint or answer. These are self-assessment activities; to check a typed answer, open Learn.

EXERCISE 1

\int e^x\cos x\,dx

Show a hint

The integral recurs after two applications.

Check my result

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

EXERCISE 2

\int e^{2x}\sin x\,dx

Show a hint

The primitive of e^(2x) includes 1/2.

Check my result

\frac{e^{2x}}5(2\sin x-\cos x)+C

EXERCISE 3

\int e^x\sin(3x)\,dx

Show a hint

Closing the cycle produces the coefficient 1+3².

Check my result

\frac{e^x}{10}(\sin(3x)-3\cos(3x))+C