Integration by Substitution

Integration is a fundamental operation in calculus that involves finding the antiderivative of a given function. One of the techniques used to evaluate integrals is called integration by substitution. This technique is also known as u-substitution, and it involves substituting a new variable for a part of the original integrand to simplify the integral.

The u-Substitution Rule

The u-substitution rule states that if we have an integral of the form:

$\int f(g(x))g'(x)dx$

Then, we can make the substitution $u = g(x)$, which gives us:

$\int f(u)du$

This form of the integral is often easier to evaluate than the original integral. We can then integrate $f(u)$ with respect to $u$ and replace $u$ with $g(x)$ to get the final result.

Example

Suppose we want to evaluate the integral:

$\int (2x+1)^2 dx$

If we expand the square, we get:

$\int (4x^2+4x+1) dx$

We can integrate each of these terms separately. The integral of $4x^2$ is $\frac{4}{3}x^3$, the integral of $4x$ is $2x^2$, and the integral of $1$ is $x$. So we have:

$\int (2x+1)^2 dx = \frac{4}{3}x^3 + 2x^2 + x + C$

Now, let's use u-substitution to evaluate the integral more efficiently. We make the substitution $u = 2x+1$, which means $du/dx = 2$. Solving for $dx$, we get $dx = du/2$. We can now rewrite the integral as:

$\int (2x+1)^2 dx = \int u^2 \frac{du}{2}$

We can integrate this expression to get:

$\int u^2 \frac{du}{2} = \frac{1}{2}\frac{u^3}{3} + C = \frac{1}{6}(2x+1)^3 + C$

This is the same result we got before, but with much less work.

Tips for Using u-Substitution

Here are some tips for using u-substitution:

1. Look for a function inside the integral that has a derivative that appears elsewhere in the integrand.
2. Make a substitution by setting the function equal to u.
3. Rewrite the integrand in terms of u.
4. Evaluate the integral in terms of u.
5. Substitute back in for the original variable.

Conclusion

Integration by substitution, or u-substitution, is a powerful tool for evaluating integrals. By making a smart substitution, we can often simplify an integral and make it easier to evaluate. With practice, you will become more comfortable with this technique and be able to use it to solve a wide variety of integrals.

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