Integration by Substitution
In this topic we shall see an important method for evaluating many complicated integrals.
Substitution for integrals corresponds to the chain rule for derivatives.
Suppose that F (u) is an antiderivative of f (u):
Assuming that u = u (x) is a differentiable function and using the chain rule, we have
Integrating both sides gives
Hence
This is the substitution rule formula for indefinite integrals.
Note that the integral on the left is expressed in terms of the variable \(x.\) The integral on the right is in terms of \(u.\)
The substitution method (also called \(u-\)substitution) is used when an integral contains some function and its derivative. In this case, we can set \(u\) equal to the function and rewrite the integral in terms of the new variable \(u.\) This makes the integral easier to solve.
Do not forget to express the final answer in terms of the original variable \(x!\)
Solved Problems
Click or tap a problem to see the solution.
Example 1
Compute the integral \[\int {{e^{\frac{x}{2}}}dx}.\]
Example 2
Find the integral \[\int {{{\left( {3x + 2} \right)}^5}dx}.\]
Example 3
Find the integral \[\int {\frac{{dx}}{{\sqrt {1 + 4x} }}}.\]
Example 4
Evaluate the integral \[\int {\frac{{xdx}}{{\sqrt {1 + {x^2}} }}}.\]
Example 5
Calculate the integral \[\int {\frac{{dx}}{{\sqrt {{a^2} - {x^2}} }}}.\]
Example 6
Evaluate the integral \[{\int {{\frac{{{x^2}}}{{{x^3} + 1}}}dx} }\] using an appropriate substitution.
Example 7
Find the integral \[\int {\sqrt[3]{{1 - 3x}}dx}.\]
Example 8
Find the integral \[\int {{\frac{{x + 1}}{{{x^2} + 2x - 5}}} dx}.\]
Example 1.
Compute the integral \[\int {{e^{\frac{x}{2}}}dx}.\]
Solution.
Let \(u = \frac{x}{2}.\) Then
Now we can easily integrate:
Example 2.
Find the integral \[\int {{{\left( {3x + 2} \right)}^5}dx}.\]
Solution.
We make the substitution \(u = 3x + 2.\) Then
The differential \(dx\) is given by
Plug all this in the integral:
Example 3.
Find the integral \[\int {\frac{{dx}}{{\sqrt {1 + 4x} }}}.\]
Solution.
We can try to use the substitution \(u = 1 + 4x.\) Hence
so
This yields
Example 4.
Evaluate the integral \[\int {\frac{{xdx}}{{\sqrt {1 + {x^2}} }}}.\]
Solution.
Let \(u = 1 + {x^2}.\) Then
We see that
Hence
Example 5.
Calculate the integral \[\int {\frac{{dx}}{{\sqrt {{a^2} - {x^2}} }}}.\]
Solution.
Let \(u = \frac{x}{a}.\) Then \(x = au,\) \(dx = adu.\) Hence, the integral is
Example 6.
Evaluate the integral \[{\int {{\frac{{{x^2}}}{{{x^3} + 1}}}dx} }\] using an appropriate substitution.
Solution.
We try the substitution \(u = {x^3} + 1.\)
Calculate the differential \(du:\)
We see from the last expression that
so we can rewrite the integral in terms of the new variable \(u:\)
Now we can easily evaluate this integral:
Express the result in terms of the variable \(x:\)
Example 7.
Find the integral \[\int {\sqrt[3]{{1 - 3x}}dx}.\]
Solution.
We use the substitution \(u = 1 - 3x.\) Then
and
After substitution we get
Example 8.
Find the integral \[\int {{\frac{{x + 1}}{{{x^2} + 2x - 5}}} dx}.\]
Solution.
We make the substitution \(u = {x^2} + 2x - 5.\) Then
or
The integral is easy to calculate with the new variable: