Calculus I
&
Engineering Mathematics 2
Workshop 7
Integration by Parts
Recall the product rule for differentiation: $\ds \frac{d}{dx}(uv)=u\,dv+v\,du.$
Re-arranging we get
Integrating both sides gives
$\ds u\,dv = \frac{d}{dx}(uv)- v\,du\;$ $\ds\Ra \int u\,dv = uv - \int v\,du.$
This method is useful when the integrand is a product of functions.
Trigonometric Substitution
This technique is used for integrals involving
$\sqrt{a^2-x^2},\ \sqrt{a^2+x^2},\ \sqrt{x^2-a^2}$.
These expressions resemble standard trigonometric identities.
We substitute $x$ using a trigonometric
function to eliminate the square root.
For example, by setting $\, x=a\sin\theta,\,$ we can rewrite the integral
$\ds \int \frac{dx}{\sqrt{a^2-x^2}}$ $\ds = \int \frac{a\cos\theta\,d\theta}{a\cos\theta}$ $\ds = \int d\theta.$
👉 $\;\sqrt{a^2-x^2} = \sqrt{a^2- a^2\sin^2\theta}$ $ = \sqrt{a^2\left(1- \sin^2\theta\right)}$ $ =\sqrt{a^2\cos^2\theta }$ $ =a\cos\theta .$
Also, since $\,x = a\sin\theta\,$ then $\,dx = a \cos\theta \,d\theta.$
Partial Fractions
This method applies to rational functions: $\ds \frac{P(x)}{Q(x)},\,$ where $\,\deg P \lt \deg Q$.
The idea is to decompose the integrand into simpler fractions.
$\ds \int $ $\ds \frac{1}{(x-1)(x+2)}$ $\ds dx$ $\ds =$ $\ds \int $ $\ds \frac{A}{x-1}$ $\ds dx$ $\ds +$ $\ds \int$ $\ds \frac{B}{x+2}$ $\ds dx\qquad\qquad \qquad $
$\qquad \qquad\quad \ds=\frac{1}{3}\int \frac{1}{x-1}\, dx - \frac{1}{3}\int \frac{1}{x+2}\,dx$
Each term can then be integrated directly.
\(A(x+2)+ B(x-1)\) \(=Ax+2A+ Bx-B\) \(=(A+B)x+2A-B\)
\( \left\{ \begin{array}{c} A+B = 0\\ 2A - B = 1 \end{array} \right. \) \( \Ra A = \dfrac{1}{3}, B = -\dfrac{1}{3} \)
Summary of Integration Techniques
Integration by parts formula: $\ds \int u\,dv = uv - \int v\,du.$
| Method | Recognise | Typical example |
|---|---|---|
| Integration by parts | $\ds \int u\,dv$ |
$\ds \int x e^x\,dx,\;$ $u = x,$ $dv= e^x\,dx$ |
| Trigonometric substitution | $\ds \sqrt{a^2-x^2},\ \sqrt{a^2+x^2}$ |
$\ds \int \frac{dx}{\sqrt{a^2-x^2}}$ $x=\sin t,$ $dx = \cos t\, dt$ |
| Partial fractions | $\ds \frac{P(x)}{Q(x)}$ | $\ds \int \frac{1}{(x-1)(x+2)}\,dx$ |