In this article, we will discuss integration involving exponential functions. But before proceeding to elaborate on how to find the integrals of exponential functions, first, let us see what is meant by integration.

Integration is the reverse process of differentiation and it is a way of adding pieces to a whole

Integration is an important concept in calculus. We use the integration to calculate area, volume, and central points.

Integration Rules

Like differentiation, there are some rules which are used to solve the problems related to integration. These rules are explained below:

  • Integration of a Constant

The integral of \int a dx is equal to ax + C.

 

  • Integration Power Rule

\int x^n dx is equal to \frac{x ^{n + 1}} {n + 1} + C

 

  • Integration Sum Rule

\int(a + b) dx is equal to \int a dx + \intb dx

 

  • Integration Difference Rule

\int(a - b) dx is equal to \int a dx - \intb dx

 

  • Multiplication by Constant

\int c f(x) dx is equal to c \int f(x) dx

 

Now, we will see what are exponential functions.

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What are Exponential Functions?

Exponential functions are the functions in which the independent variable, x, is the power or exponent of the base. The general form of exponential functions is given below:

y = a^x

Here x is the independent variable, y is the dependent variable and a is the base of the function.

Exponential and logarithmic functions are employed for modeling the growth of the population, cells in living beings, and profits. These functions are also used to model radioactive decay, consumption of resources, and depreciation over time.

Formulas for Finding Integrals of Exponential Function

Following formulas or rules are used to calculate the integrals of exponential functions:

\int e^xdx = e^x + C

\int a^x dx = \frac{a^x}{lna} + C

Now, we will solve some examples in which we will find the integrals of exponential functions.

 

Example 1

Find the integral of the following exponential function:

\int e^ {-2x} dx

Solution

We will use the substitution method to solve this example. Suppose -2x = u, then \frac{du}{dx} = -2. This means that du = -2 dx. Divide both sides of the equation by -2, so that you can have - \frac{1}{2} du = dx. Then,

= - \frac{1}{2} \int e^u du

Remember that according to the formula \int e^xdx = e^x + C.

= - \frac{1}{2}  e^u + C

Substitute u = -2x again in the above equation to get the final answer:

= -\frac{1}{2} e ^{-2x} + C

Example 2

Find the integral of the following exponential function:

\int x^2 e^{-x^3} dx

Solution

We will use substitution here to solve this example. Suppose u = -x^3. Then, \frac{du}{dx} = - 3x^2. This means that dx = - \frac{1}{3x^2} du.

= \int x^2 \cdot - \frac{1}{3x^2} e^u du

= \int - \frac{1}{3} e^u du

Take the constant before the integral sign:

= - \frac{1}{3} \int e^u du

= - \frac{1}{3} e^u + C

Substitute u = -x^3 again in the above equation to get the following final answer:

= - \frac{1}{3} e ^{-x^3} + C

 

Example 3

Find the antiderivative of the following exponential function:

\int e^x\sqrt{e^x} dx

Solution

Suppose e^x = u. This means that \frac{du}{dx} = e^x and du = e^x dx. Substitute these values in the equation below:

= \int \sqrt{u} du = \int u ^{\frac{1}{2}} du

Apply the power rule \int x^adx = \frac{x^{a + 1}} {a+ 1} to find the antiderivative of the above function:

= \frac{u ^{\frac{3}{2}}} {\frac{3}{2}}

= \frac{2}{3} u ^{\frac{3}{2}}

Substitute u = e^x again in the above equation to get:

= \frac{2}{3} e ^{\frac{3x}{2}}

 

Example 4

Find the integral of the following exponential function:

\int e^x (2e^x + 2)^2 dx

Solution

Expand the function to get the following form:

= \int 4e^{3x} + 8e^{2x} + 4e^x dx

Use the sum rule to write the above function like this:

= \int 4e^{3x} + \int 8e^{2x} + \int 4e^x dx

We will find integral of each term separately like this:

\int 4e ^{3x} dx= \frac{4}{3}e^{3x}

\int 8e ^{2x} dx= 4e^{2x}

\int 4e^x dx= 4e^x

Hence, the final answer is:

= \frac{4}{3} e ^{3x} + 4e^{2x} + 4e^x + C

 

Example 5

Find the integral of the following exponential function:

\int e^x (3e^x + 1) dx

Solution

Expand the function to get the following form:

= \int 3e^{2x} + e^x dx

Use the sum rule to write the above function like this:

= \int 3e^{2x} + \int e^{x} dx

We will find integral of each term separately like this:

\int 3e ^{2x} dx= \frac{3}{2}e^{2x}

\int e^x dx= e^x

Hence, the final answer is:

= \frac{3}{2} e ^{2x} + e^{x} + C

 

 

Example 6

Find the integral of the following exponential function:

\int \frac{e^x}{2} dx

Solution

We will use the substitution method to solve this example. Suppose x = u, then \frac{du}{dx} = 1. This means that du =  dx. Move the fraction to the left side of the integral sign like this:

=   \frac{1}{2} \int e^u du

Remember that according to the formula \int e^xdx = e^x + C.

= \frac{1}{2}  e^u + C

Substitute u = x again in the above equation to get the final answer:

= \frac{1}{2} e ^{x} + C

 

Example 7

Find the integral of the following exponential function:

\int \sqrt[3]{e^x} dx

Solution

Write the above function in radical form like this:

= \int (e^x) ^ {\frac{1}{3}} dx

= \int e ^ {\frac{x}{3}} dx

Now, use the substitution method to solve the above equation. Suppose u = \frac{x}{3}. This means that \frac{du}{dx} = \frac{1}{3}. Then, du = \frac{1}{3} dx and dx = 3 du

= \int e^u 3 du

Move the constant before the integral sign:

= 3 \int e^u du

= 3 e^u + C

Substitute u = \frac{x}{3} in the above equation again to get the following answer:

= 3 e ^ {\frac{x}{3}} + C

 

Example 8

Find the integral of the following exponential function:

\int \sqrt[4]{e^ {2x}} dx

Solution

Write the above function in radical form like this:

= \int (e^{2x}) ^ {\frac{1}{4}} dx

= \int e ^ {\frac{x}{2}} dx

Now, use the substitution method to solve the above equation. Suppose u = \frac{x}{2}. This means that \frac{du}{dx} = \frac{1}{2}. Then, du = \frac{1}{2} dx and dx = 2 du

= \int e^u 2 du

Move the constant before the integral sign:

= 2 \int e^u du

= 2  e^u + C

Substitute u = \frac{x}{2} in the above equation again to get the following answer:

= 2 e ^ {\frac{x}{2}} + C

 

Example 9

Find the integral of the following exponential function:

\int \frac{e^x} {5 e ^{2x}} dx

Solution

Take the fraction to the left side of the integral sign like this:

= \frac{1}{5} \int \frac{e^x}{e^{2x}} dx

\frac{e^x}{e^{2x}} is equal to e^{-x}:

= \frac{1}{5} \int e^{-x} dx

Now, we will use substitution to solve the above equation. Suppose u = -x, then it means that \frac{du}{dx} = -1 and du = -dx. Multiply both sides by -1 to get dx = -du.

= \frac{1}{5} - \int e^u du

= -\frac{1}{5} e^u

Substitute u = -x again in the above equation to get the final answer:

= - \frac{1}{5} e^ {-x} + C

 

Example 10

Find the integral of the following exponential function:

\int  \frac{e^x}{2^x} dx

Use the exponent rule to write the above function like this:

= \int (\frac{e}{2}) ^ x dx

Remember that \int a^x dx = \frac{a^x}{ln a}. Hence, \int (\frac{e}{2}) ^ x = \frac{ (\frac{e} {2})^x} {ln (\frac{e}{2})}.

Simplify the above equation to get the following answer:

= \frac{e^x}{(1 - ln (2)) \cdot 2^x} + C

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Rafia Shabbir