Integrating e^x, 1/x and trig functions

Standard integrals

• ∫ e^kx dx = (1/k)e^kx + c
• ∫ 1/x dx = ln|x| + c
• ∫ cos kx dx = (1/k)sin kx + c
• ∫ sin kx dx = -(1/k)cos kx + c
• ∫ sec² kx dx = (1/k)tan kx + c

Linear insides

For a linear inside ax + b, divide by a: ∫ 1/(ax + b) dx = (1/a)ln|ax + b| + c and ∫ e^{ax + b} dx = (1/a)e^{ax + b} + c. Note ∫ 1/(2x) dx = (1/2)ln|x| + c.

Definite integrals

Substitute the limits as in P2, in radians for trig. ln and e often cancel neatly: e^{2 ln 2} = 4 and ln(e²) = 2.

Worked example

Evaluate ∫₀^{ln 2} e^2x dx.

1. The integral of e^2x is (1/2)e^2x.
2. Upper limit: (1/2)e^{2 ln 2} = (1/2)(4) = 2.
3. Lower limit: (1/2)e⁰ = 1/2.

Answer: 2 - 1/2 = 3/2

Check yourself

1. Find ∫ sin 3x dx.
   • (1/3)cos 3x + c
   • -3cos 3x + c
   • (1/3)sin 3x + c
   • -(1/3)cos 3x + c
   Show answer

   -(1/3)cos 3x + c. Integrating sin gives -cos; divide by 3.

2. Find ∫ 1/(2x) dx.
   • ln|2x|/2x + c
   • (1/2)ln|x| + c
   • 2ln|x| + c
   • 1/(2x²) + c
   Show answer

   (1/2)ln|x| + c. 1/(2x) = (1/2)(1/x), which integrates to (1/2)ln|x|.

3. Evaluate ∫₀^{π/4} cos 2x dx. Give your answer as a fraction in its simplest form.
   Show answer

   1/2. The integral is (1/2)sin 2x. (1/2)sin(π/2) - (1/2)sin 0 = 1/2.

4. Evaluate ∫₁^{e²} (3/x) dx.
   Show answer

   6. The integral is 3ln x. 3ln(e²) - 3ln 1 = 6 - 0 = 6.

5. Evaluate ∫₀^{1} e^3x dx, to 3 decimal places.
   Show answer

   6.362. The integral is (1/3)e^3x. From 0 to 1: (e³ - 1)/3 = 6.362.

6. Evaluate ∫₂^{5} 1/(2x - 1) dx, to 3 decimal places.
   Show answer

   0.549. The integral is (1/2)ln|2x - 1|. (1/2)(ln 9 - ln 3) = (1/2)ln 3 = 0.549.

7. Find ∫ (e^-x + sec² x) dx.
   • e^-x + tan x + c
   • -e^-x + sec x tan x + c
   • -e^-x + tan x + c
   • -e^-x - tan x + c
   Show answer

   -e^-x + tan x + c. e^-x integrates to -e^-x; sec² x integrates to tan x.

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Sources

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

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