Section 7.1: Integration by Parts
How to integrate a product $\int u(x) v(x) dx$
Consider the product rule:
$\int \frac{d}{dx} u(x) v(x) dx = \int (u’(x)v(x)+u(x)(v’(x)) dx$
$u(x) v(x) = \int u’(x) v(x) - \int u’(x) v(x)dx$
$\int u(x) v’(x) dx = u(x) v(x) - \int u’(x) v(x) dx$
$\int u du = uv - \int v du$
Integration by Parts Formula
- need $dv$ integrable
- want $u$ simplifed by differentiation
Example 1:
$\int xe^x dx$
$u = x$ $dv = e^x dx$
$du = 1dx$ $v = e^x$
$u v - \int v du$
$= xe^x - \int e^x dx$
$= xe^x - e^x +C$
$= e^x (x-1) +C$
Example 2:
$\int ln(x) dx$
$u = lnx$ $dv = dx$
$du = \frac{1}{x} dx$ $v = x$
$u v - \int v du$
$= (lnx)x - \int x \frac{1}{x} dx$
$= (lnx)x - \int 1 dx$
$= (lnx)x - x +C$
$= x(lnx-1) +C$
Example 3:
$\int arctan(x) dx$
$u = arctanx$ $dv = dx$
$du = \frac{1}{1+x^2} dx$ $v = x$
$(arctan(x))x = - \int x \times \frac{1}{1+x^2} dx$
$r = 1 + x^2$ $dr = 2x dx$
$\frac{1}{2} \int \frac{dr}{r} = \frac{1}{2}ln \lvert r \rvert = \frac{1}{2}ln(1+x^2)$
$= x arctan(x) - \frac{1}{2} ln(1+x^2) +C$
Guide to Choose $u$, $dv$
L I A T E
<–$u$ $dv$–>
Logarithm
Inverse trig
Algebra
Trig
Exponential
Example 4:
$\int_1^{2} (xlnx) dx$
NOT DONE