Integrals — Part 2: Partial fractions, integration by parts and definite integrals: visual lesson, CBSE Class 12 Mathematics

Lesson

Integrals, Part 2

Fractions, products, and integrals with limits.

In this lesson you will

Split proper rational functions into partial fractions and integrate
Integrate products by parts, including the e^x [f + f′] pattern
Use the results for √(a² ± x²)
Evaluate definite integrals with the fundamental theorem and their properties

$y = 3x^2 + 1$

$\displaystyle\int_0^2 (3x^2 + 1)\,dx = 10$: the area between the curve, the axis and the two dashed lines.

Concept

What you will be able to do

Use partial fractions
Integrate by parts
Use $\displaystyle\int e^x[f + f']\,dx = e^xf + C$
Evaluate definite integrals
Apply the properties of definite integrals

Concept

Partial fractions

A proper rational function (numerator degree less than denominator degree) is split by the factors of its denominator. An improper one is divided first.

Divide first if the fraction is improper (numerator degree ≥ denominator degree).
Denominator factor	Partial fractions
$(x - a)(x - b)$	$\dfrac{A}{x - a} + \dfrac{B}{x - b}$
$(x - a)^2(x - b)$	$\dfrac{A}{x - a} + \dfrac{B}{(x - a)^2} + \dfrac{C}{x - b}$
$(x - a)(x^2 + bx + c)$	$\dfrac{A}{x - a} + \dfrac{Bx + C}{x^2 + bx + c}$

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