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Integration of exponential functions

How do you integrate an exponential function? This skill is important for calculating compound interest over time in finance, determining population growth models in biology, and analysing radioactive decay in physics. Use this resource to learn how.

Exponential functions have the form:
y=aekx

where a and k are constants. Examples include y=e2x+3, f(x)=e3x and y=ex1.

The antiderivative or indefinite integral of ex is ex. Therefore:

exdx=ex+c

where c is a constant.

More generally:

eax+bdx=1aeax+b+c

where a, b and c are constants.

Example 1 – integrating exponential functions

Integrate 2ex with respect to x.
2ex=2ex+c

Find e5x+1dx.

Here, a=5 and b=1.
e5x+1dx=15e5x+1+c

Find the integral 122exdx.

The limits of the integral are 1 and 2. We have:
122exdx=[2ex+c]x=1x=2=(2e2+c)(2e1+c)=2e2+c2e1c=2e22e1

Evaluate 142exdx.
142exdx=[2ex]x=1x=4=2e42e1

Evaluate 02e5x+1dx.
02e5x+1dx=[15e5x+1]x=0x=2=(15e5(2)+1)(15e5(0)+1)=(15e9)(15e1)=15e9+15e=15(ee9)

Evaluate 39ex3+4dx.
39ex3+4dx=[3ex3+4]x=3x=9=(3e93+4)(3e33+4)=(3e3+4)(3e1+4)=3(e7e3)

Exercise – integrating exponential functions

  1. Calculate the following.
    1. e3xdx
    2. e25xdx
    3. 9e3x+5e2xdx

      Hint: Divide through first.

  2. Evaluate the following.
    1. 02e3xdx
    2. 13e25xdx
    3. 119e3x+5e2xdx

    1. e3x3+c
    2. e25x5+c
    3. 9ex52e2x+c
    1. e6313
    2. 15(e7e13)
    3. 9(ee1)+52(e2e2)

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