Integration by parts
\[\int uv\,dx=u\int v\,dx-\int\left(u'\int v\,dx\right)dx\]
ILATE: Inverse trig, Log, Algebraic, Trig, Exponential.
Integrals
Integration by Parts
Understand
For a product of two functions, use integration by parts:
\[\int u\,v\,dx=u\int v\,dx-\int\left(\frac{du}{dx}\int v\,dx\right)dx.\]
Choose \(u\) (the first function) by ILATE: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential.
Example: \(\int x e^x\,dx=xe^x-\int e^x\,dx=e^x(x-1)+C\). Example: \(\int\log x\,dx=\int\log x\cdot1\,dx=x\log x-x+C\).
Special form: \(\int e^x[f(x)+f\,'(x)]\,dx=e^xf(x)+C\). E.g. \(\int e^x(\sin x+\cos x)\,dx=e^x\sin x+C\).
Also: \(\int\sqrt{a^2-x^2}\,dx=\frac x2\sqrt{a^2-x^2}+\frac{a^2}2\sin^{-1}\frac xa+C\), and similar results for \(\sqrt{x^2\pm a^2}\).
Key Concepts
- ILATE decides the first function.
- Take log x or tan⁻¹x as u with v = 1.
- Look for the eˣ[f + f′] pattern.
Formula Bank
eˣ special form
\[\int e^x\left[f(x)+f\,'(x)\right]dx=e^xf(x)+C\]
Common Mistakes
Wrong choice of u
In \(\int x e^x dx\), taking \(u=e^x\) makes the integral harder. ILATE says take \(u=x\) (algebraic before exponential).
Practice & Topic Test
Topic Practice
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Start PracticeTopic Summary
∫uv = u∫v − ∫(u′∫v). Choose u by ILATE; repeat if needed.
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