Section B.5 \(u\)-substitution

When an antiderivative is not immediately obvious, we sometimes use a substitution. Substitution is a good choice when we have one part of the integrand “looks like” the derivative of another part of the integrand.

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Let’s look at a few examples

🌌 Example 375 Compute \(\ds \int \, \cos x\sin x \ dx\).

Since the derivative of \(\cos x\) is \(-\sin x\) which "looks like" \(\sin x\text{,}\) then \(u\)-substitution (with \(u=\cos x\)) works well. Note, \(u=\sin x\) would also work.

Pick \(u\text{,}\) compute \(du\text{:}\)

\begin{align*} u \amp = \cos x \\ du \amp = -\sin x \, dx \\ -\frac{1}{\sin x}du \amp = dx \end{align*}

Rewrite integral in terms of \(u\)

\begin{align*} \int \cos x\sin x dx \amp = \int u \sin x \left[-\frac{1}{\sin x}du\right] \\ \amp = -\int u du \\ \amp = -\frac{u^2}{2} + C \\ \amp = -\frac{1}{2}\cos^2 x + C \end{align*}

🌌 Example 376 Compute \(\ds \int \frac{1}{x}\cdot \ln(15x) dx\).

Substitution is a good choice here because we have a natural log function we have \(\ds \frac{1}{x},\) which is the derivative of the natural log. So we would choose

\begin{align*} u \amp = \ln(15x) \\ du \amp = \frac{d}{dx}\Big( \ln(15x) \Big) \\ \amp = \frac{1}{15x}\cdot \frac{d}{dx}(15x) \\ \amp = \frac{1}{15x}\cdot 15dx \\ \amp = \frac{1}{x}du \end{align*}

\begin{align*} \int \frac{1}{x}\cdot \ln(15x) dx \amp = \int \ub{\ln(15x)}_{u} \cdot \ub{\frac{1}{x} dx}_{du} \\ \amp = \int udu \\ \amp = \frac{1}{2}u^2 + C \\ \amp = \frac{1}{2}( \ln(15x))^2 + C \end{align*}

We usually pause to confirm that the new integral is simpler than the original. In this case, it certainly is, so we proceed to integrate and then change back to the original variable, \(x\text{.}\)

Now you try some.

Exercises Exercises

Evaluate each of the following integrals. Use proper notation.

1.

\(\ds\int e^z \cos(e^z) dz\)

Answer.

\begin{equation*} \sin(e^z) + C \end{equation*}

2.

\(\ds\int t(t^2 - 12)^{57} dt\)

Answer.

\begin{equation*} \frac{1}{116}(t^2 - 12)^{58} + C \end{equation*}

3.

\(\ds\int 5x^2 e^{(7x^3)} dx\)

Answer.

\begin{equation*} \frac{5}{21}e^{(7x^3)} + C \end{equation*}

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