Integration – Integration by parts

Use LIATE or ALMI (priority from first to last):

• Logarithm \( (\ln x) \)
• Algebraic \( (x^n) \)
• Trigonometric \( (\sin x, \cos x) \)
• Exponential \( (e^x, a^x) \)

Choose \(u\) from the top, and \(dv\) is the rest.

Example 1: \( \int x e^x\,dx \)

Let \(u = x\), \(dv = e^x dx\). Then \(du = 1dx\), \(v = e^x\). \[ \int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C = e^x(x - 1) + C \]

Example 2: \( \int x\sin x\,dx \)

Let \(u = x\), \(dv = \sin x dx\). Then \(du = dx\), \(v = -\cos x\). \[ \int x\sin x dx = -x\cos x + \int \cos x\,dx = -x\cos x + \sin x + C \]

Example 3: \( \int x\ln x\,dx \)

Important: \(\ln x\) must be chosen as \(u\). Let \(u = \ln x\), \(dv = x\,dx\). Then \(du = \frac{1}{x}dx\), \(v = \frac{x^2}{2}\). \[ \int x\ln x\,dx = \frac{x^2}{2}\ln x - \frac{x^2}{4} + C \]

Example 4 (Repeated): \( \int x^2 e^x\,dx \)

First apply parts with \(u = x^2\). You must apply it twice: \[ \int x^2 e^x dx = x^2 e^x - 2 \int x e^x dx \] Then use the result from Example 1. Final answer: \[ \int x^2 e^x dx = e^x(x^2 - 2x + 2) + C \]

Example 5 (Exam Style): \( \int x e^{-x}\,dx \)

Let \(u = x\), \(dv = e^{-x} dx\). Then \(du = dx\), \(v = -e^{-x}\). \[ \int x e^{-x} dx = -x e^{-x} - \int (-e^{-x})dx = -x e^{-x} - e^{-x} + C = -(x + 1)e^{-x} + C \]

• If \(\ln x\) is present, it is ALWAYS \(u\).
• If a power of \(x\) is multiplied by \(e^x\) or \(\sin x\), integration by parts is needed.
• Expect repeated integration if \(x^2\) or higher powers appear.
• Don’t forget + C for indefinite integrals (−1 mark often!).
• For definite integrals, apply limits AFTER finishing parts.