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Special Integrals — Part 2: Quadratics, roots, trigonometric forms

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Lesson

Special Integrals

Standard results for quadratics, roots, trigonometric forms and a few special integrands.

In this lesson you will

Integrate reciprocals of quadratics and their square roots by completing the square
Split a linear numerator into a multiple of the derivative and a constant
Integrate square roots of quadratics
Use the half-angle substitution and the A plus B split for trigonometric forms
Evaluate the special integrals of one plus or minus x squared over one plus x to the fourth, root tan x and root cot x

Part 2 at a glance.

Concept

Five standard forms

Sum and differences of squares
Reciprocals of square roots
Learn the factor in front of each

Learn these five; every quadratic form reduces to one of them.
Integrand	Integral
$\dfrac{1}{x^2 + a^2}$	$\dfrac1a\tan^{-1}\dfrac{x}{a} + C$
$\dfrac{1}{x^2 - a^2}$	$\dfrac{1}{2a}\log\left\|\dfrac{x - a}{x + a}\right\| + C$
$\dfrac{1}{a^2 - x^2}$	$\dfrac{1}{2a}\log\left\|\dfrac{a + x}{a - x}\right\| + C$
$\dfrac{1}{\sqrt{a^2 - x^2}}$	$\sin^{-1}\dfrac{x}{a} + C$
$\dfrac{1}{\sqrt{x^2 \pm a^2}}$	$\log\left\|x + \sqrt{x^2 \pm a^2}\right\| + C$

Concept

Completing the square

Any quadratic $ax^2 + bx + c$ can be written as $a\left[(x + h)^2 \pm k^2\right]$ .

Halve the coefficient of $x$, square it, add and subtract.
Quadratic	Completed square	Form reached
$x^2 + 6x + 13$	$(x + 3)^2 + 2^2$	$u^2 + a^2$
$x^2 - 4x + 13$	$(x - 2)^2 + 3^2$	$u^2 + a^2$
$8 + 2x - x^2$	$3^2 - (x - 1)^2$	$a^2 - u^2$
$2x^2 + 8x + 10$	$2\left[(x + 2)^2 + 1\right]$	take out the 2 first