All subjects · Further Mathematics · WFM03 Further Pure Mathematics 3

Integrating hyperbolic functions

Edexcel IAL Further Mathematics · WFM03 Further Pure Mathematics 3

Reverse the derivatives to integrate sinh, cosh and sech². For powers, use the double angle identities or the substitution trick for odd powers, just as in trigonometry.

Practise this in the app: instant marking and a fix for every mistake. Free, no email needed to start.

Practise this lesson

Standard integrals

∫ sinh x dx = cosh x + c
∫ cosh x dx = sinh x + c
∫ sech² x dx = tanh x + c
∫ tanh x dx = ln(cosh x) + c
∫ sinh(ax) dx = (1/a) cosh(ax) + c

No sign changes: integrating sinh gives +cosh.

Squares: double angle identities

cosh² x = (cosh 2x + 1)/2 and sinh² x = (cosh 2x - 1)/2.

So ∫ cosh² x dx = (1/4) sinh 2x + x/2 + c and ∫ sinh² x dx = (1/4) sinh 2x - x/2 + c.

Odd powers and exponentials

For sinh³ x, write sinh³ x = sinh x(cosh² x - 1); with u = cosh x this integrates to cosh³ x/3 - cosh x + c.

You can always fall back on the exponential definitions, for example ∫ e^x cosh x dx = ∫ (e^2x + 1)/2 dx.

Worked example

Find the exact value of ∫ from 0 to ln 3 of cosh² x dx.

cosh² x = (cosh 2x + 1)/2, so the integral is [sinh 2x/4 + x/2]
sinh(2 ln 3) = sinh(ln 9) = (9 - 1/9)/2 = 40/9
Value = (40/9)/4 + (ln 3)/2

Answer: 10/9 + (1/2) ln 3, about 1.660

Key idea: ∫ sinh = cosh, ∫ cosh = sinh, ∫ sech² = tanh, ∫ tanh = ln cosh. Use cosh 2x identities for squares and cosh² - sinh² = 1 for odd powers.

Check yourself

Evaluate ∫ from 0 to 1 of cosh x dx, to 3 decimal places.
Show answer
1.175. [sinh x] from 0 to 1 = sinh 1 = 1.175.
Find the exact value of ∫ from 0 to ln 2 of sinh x dx. Give your answer as a fraction in its simplest form.
Show answer
1/4. [cosh x] = cosh(ln 2) - cosh 0 = 5/4 - 1 = 1/4.
Which is ∫ sinh² x dx?
- (1/4) sinh 2x + x/2 + c
- (1/3) sinh³ x + c
- (1/4) sinh 2x - x/2 + c
- x/2 - (1/4) sinh 2x + c
Show answer
(1/4) sinh 2x - x/2 + c. sinh² x = (cosh 2x - 1)/2, which integrates to sinh 2x/4 - x/2.
Evaluate ∫ from 0 to 1 of tanh x dx, to 3 decimal places.
Show answer
0.434. [ln cosh x] from 0 to 1 = ln(cosh 1) = 0.434.
Evaluate ∫ from 0 to ln 3 of cosh² x dx, to 3 decimal places.
Show answer
1.660. 10/9 + (1/2) ln 3 = 1.660.
Which is ∫ sech² 3x dx?
- 3 tanh 3x + c
- (1/3) tanh 3x + c
- tanh 3x + c
- -(1/3) tanh 3x + c
Show answer
(1/3) tanh 3x + c. d/dx tanh 3x = 3 sech² 3x, so divide by 3.
Evaluate ∫ from 0 to 1 of sinh³ x dx, to 3 decimal places.
Show answer
0.348. sinh³ x = sinh x(cosh² x - 1), giving [cosh³ x/3 - cosh x] from 0 to 1 = 0.348.

Practise with instant feedback

Sources

Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP3

Last updated 8 November 2026. Spotted a mistake? Tell us.