u-substitution

When an integral can not be calculated using the standard integration rules introduced in an earlier article, another method called u-substitution can be used. U-substitution works by recognizing that an integral is of a particular form, after which the integral can be solved using a step-by-step method.

Step-by-step method

U-substitution can be applied to integrals that are of the following form.

With $f(x)\;and\;g(x)$ being functions, $g'(x)$ the derivative of $g(x)$ and $a$ a constant that is multiplied with $g'(x)$ This form is not always easily recognized thus some integrals may need rewriting in order to recognize this form.

The form can be explained as the integral of a function $f$, with another function $g$ as an argument of $f$ multiplied with the derivative of $g$ and a constant.

Assuming the integral is of the form as explained above, the steps for applying u-substitution are as follows:

1. Let $g(x) = u$

2. Differentiate both sides over x and $\frac{du}{dx} = g'(x)$ is obtained

3. Which is the same as $du = g'(x)dx$

4. Substitute this in the integral and a new integral $\int{}{}udu$ is obtained

5. After integration by using the standard rules, $\frac{1}{2}u^{2}+c$ is found

6. The last step is replacing $u$ by $g(x)$, thus the final answer equals $\frac{1}{2}g(x)^{2} + c$

Example

The integral below can not be solved using the standard integration rules, thus u-substitution will be applied in order to still solve this integral.

The form on which u-substitution can be applied is recognized.

In this case $f(g(x))=cos(g(x)),\;g(x)=x^{3}+4$ and $ag'(x)=6x^{2}$

The solution is worked out in the steps below:

1. $x^{3}+4 = u$ (Substitution)
2. $\frac{du}{dx} = 3x^{2}$ (Differentiation)
3. $du = 3x^{2}dx$ (Rewriting)
4. $\int{}{}\cos(u)\frac{6x^{2}}{3x^{2}}du$ (Substitution in integral)
5. $2\int{}{}\cos(u)du$ (Rewriting)
6. $2\sin(u)+c$ (Integration)
7. $2\sin(x^{3}+4) + c$ (Substitution of $g(x)$ in u)
8. $\int{}{}6x^{2}\cos(x^{3}+4)dx = 2\sin(x^{3}+4) + c$ (Solution)

Now is shown how to apply u-substitution to integrals that can not be solved by solely using the standard rules for integration. In the next article will be explained how to solve integrals that can not be solved using the standard rules or u-substitution by using another method called Integration by Parts.