IB AA SL: Core and Optional Content Core: Sine/cosine-square integrals using cosine double angle. Op

Question

IB AA SL: Core and Optional Content

Core: Sine/cosine-square integrals using cosine double angle.
Optional: Formulae that assume sec notation.

Use identities to remove the square, then integrate in radians.

$\cos^2x=\dfrac{1+\cos2x}2\qquad\sin^2x=\dfrac{1-\cos2x}2$

$\tan^2x=\sec^2x-1$

Integrand	Integral
$\cos^2x$
$\sin^2x$
$\tan^2x$

Example

Integrate $\sin^2(5x+1)$ with respect to $x$.

$\int\sin^2(5x+1)\,\mathrm{d}x$${}=\int\dfrac{1-\cos(2(5x+1))}{2}\,\mathrm{d}x$${}=\dfrac12\int[1-\cos(10x+2)]\,\mathrm{d}x$${}=\dfrac12\left[x-\dfrac{\sin(10x+2)}{10}\right]+C$${}=\dfrac{x}{2}-\dfrac{\sin(10x+2)}{20}+C$

Accepted Answer

$\displaystyle\int\cos^2x\,\mathrm{d}x=\dfrac{x}{2}+\dfrac{\sin2x}{4}+C$, $\displaystyle\int\sin^2x\,\mathrm{d}x=\dfrac{x}{2}-\dfrac{\sin2x}{4}+C$, $\displaystyle\int\tan^2x\,\mathrm{d}x=\tan x-x+C$

Integrand	Integral
\(\cos^2x\)
\(\sin^2x\)
\(\tan^2x\)

Integration of Quadratic Trigonometric Functions

Integration of Quadratic Trigonometric Functions

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