Integration of Exponential Functions

Expert reviewed • 21 July 2024 • 3 minute read

HSC Maths Advanced Syllabus

establish and use the formulae $\int e^xdx=e^x+C$ and $\int e^{ax+b}dx = \frac{1}{a}e^{ax+b}+C$

Note:

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Forms of Integration With Exponential Functions

Similar to all concepts of integration we have learnt so far, integrating exponential functions can also be completed using specific formulas. As such, specific expressions must be manipulated to fit the formulas provided. The formulas are as follows:

\int e^xdx=e^x+C \quad and \quad \int e^{ax+b}dx=\frac{1}{a}e^{ax+b}+C

Practice Question 1

Evaluate the following expression, using both the formulas above and prior knowledge: $\int_0^54e^{2x+3}dx$

We can see that the expression is a definite integral, meaning we must use formulas provided to solve for a numerical value.

Applying the formulas above:

\int_0^54e^{2x+3}dx=4[\frac{1}{2}e^{2x+3}]_0^5\\=4(\frac{e^{2(5)+3}}{2}-\frac{e^{2(0)+3}}{2})\\=4(\frac{e^{13}-e^3}{2})\\=2(e^{13}-e^3)\quad or \quad 2e^3(e^{10}-1)

Differentiation of Exponential and Logarithmic Functions

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Integration of the Reciprocal Function

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