07 4antitrig

Integration of sin(x), cos(x)

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07.4rules.gif

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Note: Take extra care over when to use the negative sign.

  • The derivative of $\cos \big( x \big) \text{ is } -\sin \big( x \big)$
  • The antiderivative of $\sin \big( x \big) \text{ is } -\cos \big( x \big)$

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Example 1

Find
a) .. $\displaystyle{ \int } \sin \big(3x \big) \; dx$

b) .. $\displaystyle{ \int } 4\cos \left( \frac{x}{2} \right) \; dx$

Solution

a) .. $\displaystyle{ \int } \sin \big( 3x \big) \; dx$
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… … … … $a = 3 \qquad \displaystyle{ \int } \sin \big( ax \big) \; dx = -\dfrac{1}{a}\cos \big( ax \big) + c$
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… … … $\displaystyle{ \int } \sin \big( 3x \big) \; dx = -\dfrac{1}{3}\cos \big( 3x \big) + c$

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b) .. $\displaystyle{ \int } 4\cos \left( \frac{x}{2} \right) \; dx$
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… … … … $a= \dfrac{1}{2} \qquad \displaystyle{ \int } \cos \big( ax \big) \; dx = \dfrac{1}{a}\sin \big( ax \big) + c$
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… … … $\displaystyle{ \int } 4\cos \left( \frac{x}{2} \right) \; dx = 8\sin \left( \dfrac{x}{2} \right) + c$

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Note: Antiderivative involving tan or giving tan as the solution is not part of the Methods Course.

… … $\displaystyle{ \int } \dfrac{1}{\cos^2 \big( x \big)} \; dx = \tan \big( x \big) + c$

… … $\displaystyle{ \int } \dfrac{a}{\cos^2 \big( ax \big)} \; dx = \tan \big( ax \big) + c$

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