Integration by parts

The formula

∫ u (dv/dx) dx = uv - ∫ v (du/dx) dx.

Choose u to be the part that becomes simpler when differentiated; dv/dx must be something you can integrate.

Choosing u

• A polynomial times e^x or a trig function: u = the polynomial.
• Anything with ln x: u = ln x (you cannot easily integrate ln x).

∫ ln x dx: write it as ∫ ln x × 1 dx, with u = ln x and dv/dx = 1. The result is x ln x - x + c.

Repeated parts

• x² e^x needs parts twice: each time the power of x drops.
• e^x sin x: after two applications the original integral reappears. Call it I and solve for I.

Worked example

Find ∫ x e^2x dx.

1. u = x, du/dx = 1; dv/dx = e^2x, v = (1/2)e^2x
2. ∫ = (1/2)x e^2x - ∫ (1/2)e^2x dx

Answer: (1/2)x e^2x - (1/4)e^2x + c

Check yourself

1. Find ∫ x cos x dx.
   • x sin x + cos x + c
   • x sin x - cos x + c
   • (x²/2)sin x + c
   • -x sin x + cos x + c
   Show answer

   x sin x + cos x + c. u = x, v = sin x: x sin x - ∫ sin x dx = x sin x + cos x + c.

2. For ∫ x² ln x dx, which choice of u should you make?
   • u = x²
   • u = x² ln x
   • u = ln x
   • u = 1
   Show answer

   u = ln x. ln x is hard to integrate but easy to differentiate.

3. Evaluate ∫₁^{e} ln x dx.
   Show answer

   1. ∫ ln x dx = x ln x - x. From 1 to e: (e - e) - (0 - 1) = 1.

4. Use integration by parts to find ∫₀^{2} x e^x dx, to 3 decimal places.
   Show answer

   8.389. ∫ x e^x dx = x e^x - e^x. From 0 to 2: (2e² - e²) - (0 - 1) = e² + 1 = 8.389.

5. Show that ∫₀^{π} x sin x dx = π, and give the value to 2 decimal places.
   Show answer

   3.14. ∫ x sin x dx = -x cos x + sin x. From 0 to π: (-π(-1) + 0) - 0 = π = 3.14.

6. Integrating by parts twice, find ∫₀^{1} x² e^x dx to 3 decimal places.
   Show answer

   0.718. Using parts twice: ∫ x² e^x dx = e^x(x² - 2x + 2). From 0 to 1: e(1) - 2 = 0.718.

7. Find ∫ e^x sin x dx.
   • e^x(sin x - cos x) + c
   • (1/2)e^x(sin x - cos x) + c
   • (1/2)e^x(sin x + cos x) + c
   • -e^x cos x + c
   Show answer

   (1/2)e^x(sin x - cos x) + c. Parts twice gives I = e^x sin x - e^x cos x - I, so 2I = e^x(sin x - cos x).

8. With u = ln x, find the value of ∫₁^{2} x ln x dx to 3 decimal places.
   Show answer

   0.636. u = ln x, v = x²/2: (x²/2)ln x - x²/4. From 1 to 2: (2ln 2 - 1) - (0 - 1/4) = 2ln 2 - 3/4 = 0.636.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 4 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.