Business Mathematics and Statistics · Ch 2 — Integral Calculus – I (Indefinite/Definite Integrals)

Integration by Substitution and by Parts

Integration by substitution

When the integrand is a function of a simpler expression (like $(2x+3)^5$ ), substituting $u$ for that inner expression often reduces the integral to a standard form. If $u=g(x)$ , then $du=g'(x)\,dx$ , and

\int f(g(x))\,g'(x)\,dx=\int f(u)\,du

For example, for $\int(2x+3)^5\,dx$ , let $u=2x+3$ , so $du=2\,dx$ , i.e. $dx=\frac{du}{2}$ :

\int(2x+3)^5\,dx=\int u^5\cdot\frac{du}{2}=\frac{1}{2}\cdot\frac{u^6}{6}+C=\frac{(2x+3)^6}{12}+C

Integration by parts

For a PRODUCT of two functions, integration by parts uses

\int u\,dv=uv-\int v\,du

Choosing which factor is $u$ (differentiated) and which is $dv$ (integrated) is guided by the ILATE rule — prefer, in order, Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential functions as $u$ , since these get SIMPLER when differentiated.

Note

A poor choice of $u$ makes the integral harder, not easier …

Definition 1Integration by Substitution

Replacing an inner expression $g(x)$ with a new variable $u$ so that $\int f(g(x))g'(x)\,dx$ becomes the standard-form int …

Definition 2Integration by Parts

$\int u\,dv=uv-\int v\,du$ , used for integrating a product of two functions; $u$ is chosen (via the ILATE rule) so that $du …

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