Porisma

Techniques of integration

Learn one idea at a time

Choose a unit. Each has worked examples and practice.

  1. Recognise standard forms

    Match the inner derivative and retain the interval conditions.

  2. Integrate rational functions

    Choose logarithmic forms or partial fractions from the denominator.

  3. Use trigonometric identities

    Rewrite products and powers before integrating.

  4. Change the variable

    Replace both the integrand and differential, then return to the original variable.

  5. Choose and check integration by parts

    Choose the factor to differentiate and check whether the remaining integral is simpler.

Read or print the complete chapter

By the end, you can

  • Recognise reverse chain-rule patterns
  • Use substitution and integration by parts
  • Check antiderivatives by differentiating

5 focused units · about 35 minutes for the full chapter. Allow extra time for practice.

The central idea

The printed outline covers f′(x)[f(x)]nf^{\prime}(x)[f(x)]^n (including n=−1n=-1), f′(x)ef(x)f^{\prime}(x)e^{f(x)}, sin⁡2x\sin^2x, cos⁡2x\cos^2x, tan⁡2x\tan^2x, 1/(a2+x2)1/(a^2+x^2), 1/a2−x21/\sqrt{a^2-x^2}, 1/(a2−x2)1/(a^2-x^2) and 1/(x2−a2)1/(x^2-a^2), followed by a given substitution and integration by parts. The methods below connect these forms to the worked examples and practice routes.

Open the chapter exploration

Worked example

∫2x(x2+1)3 dx\int2x(x^2+1)^3\,dx: let u=x2+1u=x^2+1, so du=2x dxdu=2x\,dx. The result is u4/4+C=(x2+1)4/4+Cu^4/4+C=(x^2+1)^4/4+C.

Substituting the expression but leaving the old differential mixes variables.

Check your understanding

Using the given substitution u=x2+1u=x^2+1, find ∫xln⁡(x2+1) dx\int x\ln(x^2+1)\,dx. Explain the next method needed after substitution, give the real domain and verify the final answer by differentiation.

Full mastery checklist
Complete source practice map

Practice questions

55 examples and practice questions, in 10 sections.

More practice

Explain why your chosen method simplifies the integral, track the variable and domain, and differentiate the final expression before accepting it.

Ready to try this on your own?

For “Techniques of integration”, try a question without the working and explain your method. This records your confidence, saved in this browser only.

Ready to practise?Chapter practice