Integration of Exponential and Trigonometric Functions

Exponential Integration Formula

Here is the exponential integration formula supplied by the formula sheet.

$\int{e^{ax}}dx=\frac{1}{x}e^{ax}+C$

Here we simply need to divide the function by the derivative of what e is to the power of. This is again the reverse of the exponential differentiation process and differentiating the integral is a good way to check our result.

Exponential Integration Example

$int{e^{-x}+e^{x}}dx \newline =\frac{1}{-1}e^{-x}+\frac{1}{1}e^{x}+C \newline =-e^{-x}+e^{x}+C$

Don’t forget to include the arbitrary constant c.

Trigonometric Integration Formula

The Trigonometric integration methods that we use are also given in the formula sheet.

The integral of sin is just negative cos divided by the derivative of whatever is in the brackets

$\int{sin(ax)}dx=-\frac{1}{a}cos(ax)$

The integral of cos is just sin divided by the derivative of whatever is in the brackets

$\int{cos(ax)}dx=\frac{1}{a}sin(ax)$

You are not required to calculate the integral of tan(x).

Trigonometric Integration Example

$int{cos(2x+1)}dx$

Using the formula above we can see that a is equal to 2.

$\int{cos(2x+1)}dx \newline =\frac{1}{\frac{d}{dx}(2x+1)}sin(2x+1) \newline =\frac{1}{2}sin(2x+1) +C$

Remember to always add in the arbitrary constant!

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