IB AA SL: core and optional content Core: integration involving sine and cosine and the prescribed r

Question

IB AA SL: core and optional content

Core: integration involving sine and cosine and the prescribed reverse-chain forms. Optional: formulae that assume sec notation.

Angles are in radians; $a\ne0$. Use intervals where the integrand is defined.

Integration reverses the following derivatives.

$\dfrac{\mathrm{d}}{\mathrm{d}x}\sin x=\cos x\qquad\int\cos x\,\mathrm{d}x=\sin x+C$

$\dfrac{\mathrm{d}}{\mathrm{d}x}\cos x=-\sin x\qquad\int\sin x\,\mathrm{d}x=-\cos x+C$

$\dfrac{\mathrm{d}}{\mathrm{d}x}\tan x=\sec^2x\qquad\int\sec^2x\,\mathrm{d}x=\tan x+C$

Linear arguments

$\int\sin(ax+b)\,\mathrm dx=-\dfrac{\cos(ax+b)}a+C$

$\int\cos(ax+b)\,\mathrm dx=\dfrac{\sin(ax+b)}a+C$

$\int\sec^2(ax+b)\,\mathrm dx=\dfrac{\tan(ax+b)}a+C$

Sine integrates to negative cosine. If a = 0, integrate the resulting constant when defined.

Accepted Answer

$\displaystyle\int\cos x\,\mathrm{d}x=\sin x+C$, $\displaystyle\int\sin x\,\mathrm{d}x=-\cos x+C$, $\displaystyle\int\sec^2x\,\mathrm{d}x=\tan x+C$; replace $x$ by $ax+b$ and divide by $a$

Integration of Trigonometric Functions

Integration of Trigonometric Functions

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