Mastering Integration by Substitution for Your Mathematics Exam

What the examiner is testing

The examiner is assessing your ability to transform a complex integral into a simpler one using a given or chosen substitution. Marks are typically awarded for correctly differentiating the substitution, rewriting the entire integral in terms of the new variable, and then integrating the simplified expression.

The method

1. Identify the substitution: This is often given in the question (e.g., "use the substitution \(u = \sin x\)"). If not, look for a part of the integrand whose derivative is also present (or a constant multiple of it).
2. Differentiate the substitution: Find \(\frac{du}{dx}\) (or \(\frac{du}{dt}\), etc.).
3. Rearrange for \(dx\): Express \(dx\) in terms of \(du\) and the original variable (e.g., \(dx = \frac{du}{du/dx}\)).
4. Substitute into the integral: Replace every instance of the original variable and \(dx\) with their \(u\) equivalents. Ensure the entire integral is now in terms of \(u\).
5. Change the limits (for definite integrals): If it's a definite integral, substitute the original limits into your substitution equation to find the new limits in terms of \(u\).
6. Integrate with respect to \(u\): Solve the new, simpler integral.
7. Substitute back (for indefinite integrals): If it was an indefinite integral, replace \(u\) with its original expression in terms of \(x\) (or \(t\), etc.) and add the constant of integration, \(+C\).

Worked example

Integrate \(\int (2x+1)^3 \, dx\).

Let \(u = 2x+1\).

Then \(\frac{du}{dx} = 2\).

Rearranging for \(dx\): \(dx = \frac{du}{2}\).

Substitute into the integral:
$$ \int u^3 \left(\frac{du}{2}\right) $$
$$ = \frac{1}{2} \int u^3 \, du $$
Integrate with respect to \(u\):
$$ = \frac{1}{2} \left( \frac{u^4}{4} \right) + C $$
$$ = \frac{u^4}{8} + C $$
Substitute back \(u = 2x+1\):
$$ = \frac{(2x+1)^4}{8} + C $$

Sanity check: Differentiate the result.
\(\frac{d}{dx} \left( \frac{(2x+1)^4}{8} + C \right) = \frac{1}{8} \times 4(2x+1)^3 \times 2 = (2x+1)^3\). This matches the original integrand.

Worked example: a harder one

Evaluate \(\int_0^1 x e^{x^2+1} \, dx\).

First attempt: Let \(u = x^2+1\).
Then \(\frac{du}{dx} = 2x\).
So \(dx = \frac{du}{2x}\).

Substituting into the integral:
$$ \int_0^1 x e^u \left(\frac{du}{2x}\right) $$
$$ = \int_0^1 \frac{1}{2} e^u \, du $$
Notice how the \(x\) cancelled out. This is a good sign that the substitution was appropriate.

Now, change the limits:
When \(x=0\), \(u = 0^2+1 = 1\).
When \(x=1\), \(u = 1^2+1 = 2\).

So the integral becomes:
$$ \int_1^2 \frac{1}{2} e^u \, du $$
Integrate with respect to \(u\):
$$ = \left[ \frac{1}{2} e^u \right]_1^2 $$
Apply the limits:
$$ = \frac{1}{2} e^2 - \frac{1}{2} e^1 $$
$$ = \frac{1}{2} (e^2 - e) $$

Why the obvious first move might fail: If we had chosen \(u=x\), then \(\frac{du}{dx}=1\), \(dx=du\), and the integral becomes \(\int_0^1 u e^{u^2+1} \, du\), which is the same integral we started with and doesn't simplify anything. The key is to choose a substitution where \(\frac{du}{dx}\) (or a multiple of it) is also present in the integrand, allowing terms to cancel or simplify.

Practice

1. \(\int (3x-2)^4 \, dx\)
2. \(\int x \sqrt{x^2+5} \, dx\)
3. \(\int_0^{\frac{\pi}{2}} \sin x \cos^3 x \, dx\)
4. \(\int \frac{e^{2x}}{e^{2x}+3} \, dx\)

Answers:
1. \(\frac{(3x-2)^5}{15} + C\)
2. \(\frac{1}{3}(x^2+5)^{3/2} + C\)
3. \(\frac{1}{4}\)
4. Working for Q4:
Let \(u = e^{2x}+3\).
Then \(\frac{du}{dx} = 2e^{2x}\).
Rearranging for \(dx\): \(dx = \frac{du}{2e^{2x}}\).
Substitute into the integral:
$$ \int \frac{e^{2x}}{u} \left(\frac{du}{2e^{2x}}\right) $$
$$ = \int \frac{1}{2u} \, du $$
$$ = \frac{1}{2} \int \frac{1}{u} \, du $$
Integrate with respect to \(u\):
$$ = \frac{1}{2} \ln|u| + C $$
Substitute back \(u = e^{2x}+3\):
$$ = \frac{1}{2} \ln|e^{2x}+3| + C $$
Since \(e^{2x}+3\) is always positive, the absolute value is not strictly necessary:
$$ = \frac{1}{2} \ln(e^{2x}+3) + C $$

The three mistakes that lose marks

1. Not changing \(dx\): Forgetting to replace \(dx\) with its equivalent in terms of \(du\) and the original variable. This often leads to an integral that still contains both \(x\) and \(u\), which cannot be solved.
   • Wrong answer example: For \(\int (2x+1)^3 \, dx\) with \(u=2x+1\), if you just write \(\int u^3 \, dx\), you're stuck.
2. Not changing the limits for definite integrals: Applying the original limits to the integral in terms of \(u\). This results in an incorrect numerical value.
   • Wrong answer example: For \(\int_0^1 x e^{x^2+1} \, dx\) with \(u=x^2+1\), if you calculate \(\left[ \frac{1}{2} e^u \right]_0^1\), you get \(\frac{1}{2}(e^1 - e^0) = \frac{1}{2}(e-1)\), which is incorrect.
3. Forgetting to substitute back (for indefinite integrals): Leaving the final answer in terms of \(u\) instead of the original variable \(x\). The question asks for an integral with respect to \(x\), so the answer must be in terms of \(x\).
   • Wrong answer example: For \(\int (2x+1)^3 \, dx\), if you give \(\frac{u^4}{8} + C\) as the final answer, you will lose marks.

30-second recap

Integration by substitution simplifies integrals by replacing a complex part with a new variable \(u\). Remember to differentiate your substitution to find \(dx\) in terms of \(du\), change limits for definite integrals, and substitute back for indefinite integrals. The goal is to transform the entire integral into the new variable.