Primitive of Cosine Function

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Theorem

$\ds \int \cos x \rd x = \sin x + C$

where $C$ is an arbitrary constant.


Corollary

$\ds \int \cos a x \rd x = \frac {\sin a x} a + C$


Corollary 2

$\ds \int \map \cos {a x + b} \rd x = \frac {\map \sin {a x + b} } a + C$


Proof 1

From Derivative of Sine Function:

$\dfrac \d {\d x} \sin x = \cos x$

The result follows from the definition of primitive.

$\blacksquare$


Proof 2

\(\ds \int \cos x \rd x\) \(=\) \(\ds \frac 1 2 \int \paren {e^{i x} + e^{-i x} } \rd x\) Euler's Cosine Identity
\(\ds \) \(=\) \(\ds \frac 1 {2 i} \paren {e^{i x} - e^{-i x} } + C\) Primitive of Exponential of a x
\(\ds \) \(=\) \(\ds \sin x + C\) Euler's Sine Identity

$\blacksquare$


Also see


Sources

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