Integrating Trigonometric Functions

We can recall that:

\[ \textrm{If }y=\sin(ax)\textrm{ then }\frac{dy}{dx}=a\cos(ax) \]

\[ \textrm{If }y=\cos(ax)\textrm{ then }\frac{dy}{dx}=-a\sin(ax) \]

When integrating, we do the opposite of differentiation, so:

\[ \int\cos(ax)\;dx = \frac{\sin(ax)}{a} + C \]

\[ \int\sin(ax)\;dx = \frac{-\cos(ax)}{a} + C \]

Example

Find the following:

1. \(\int\cos(4x)\;dx\)

2. \(\int 6\sin(3x)\;dx\)

3. \(\int\left(5\cos(-x) + \sin(3x)\right)\;dx\)

Solution:

1. Using the rule \(\int\cos(ax)\;dx = \frac{\sin(ax)}{a} + C\), we can find:

   \[ \int\cos(4x)\;dx = \frac{\sin(4x)}{4} + C \]

2. First, we should note \(\int 6\sin(3x)\;dx= 6\int \sin(3x)\;dx\). Now we can apply the rules above to get:

   \[ 6\int \sin(3x)\;dx = 6\times \frac{-\cos(3x)}{3} + C = -2\cos(3x) \]

3. Splitting up the integral and extracting constants gives:

   \[\begin{split} \begin{aligned} \int \left(5\cos(-x) + \sin(3x)\right)\;dx & = 5\int \cos(-x)\;dx + \int \sin(3x)\;dx \\ & 5\times \frac{\sin(-x)}{-1} + \frac{-\cos(3x)}{3} + C \\ & -5 \sin(-x) - \frac{\cos(3x)}{3} + C \end{aligned} \end{split}\]

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As one would, hopefully, expect, `sympy` can handle these operations.

from sympy import symbols, sin, cos, integrate

x = symbols('x')

integrate(cos(4 * x))

\[\displaystyle \frac{\sin{\left(4 x \right)}}{4}\]

integrate(6 * sin(3 * x))

\[\displaystyle - 2 \cos{\left(3 x \right)}\]

Python Note

sympy performs an additional simplification of \(-5\sin(-x) = 5\sin(x)\).

integrate(5 * cos(-1 * x) + sin(3 * x))

\[\displaystyle 5 \sin{\left(x \right)} - \frac{\cos{\left(3 x \right)}}{3}\]

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