Topic 6.9 Notes – Integrating Using Substitution

1. What U-Substitution Is

You already know the Chain Rule:

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

U‑substitution is the reverse idea. If a derivative looks like
$f'(g(x)) \cdot g'(x)$, then its antiderivative is $f(g(x)) + C$.

So when integrating, you’re hunting for this pattern:

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

If you let
$u = g(x)$,
then $du = g'(x)\,dx$,

and the integral becomes

$$\int f'(u)\,du$$

Much simpler.

Here’s the structure visually. Notice how the inner function $g(x)$ shows up inside the composite and its derivative $g'(x)$ appears as a separate factor:

Chain rule structure for a composite function

The key idea is recognizing the “inside function” and seeing whether its derivative is also present.

2. When to Use Substitution

You’ll use substitution when:

• There’s a function inside another function
• The derivative of the inside is present (or almost present)

Common patterns:

• $\int (g(x))^n g'(x)\,dx$
• $\int \frac{g'(x)}{g(x)}\,dx$
• $\int e^{g(x)} g'(x)\,dx$
• $\int \sin(g(x)) g'(x)\,dx$
• $\int \cos(g(x)) g'(x)\,dx$

Example:

$$\int 3x^2 (x^3 + 5)^4 dx$$

You should notice:

• Inside function → $x^3 + 5$
• Its derivative → $3x^2$

That’s your signal.

If the derivative is off by a constant, that’s fine. You just factor the constant out and adjust.

A quick mental check:
“If I let this inside expression equal $u$, does the rest turn into $du$?”

3. The U-Substitution Process

Indefinite Integrals

Suppose we evaluate:

$$\int 2x e^{x^2} dx$$

1. Let $u = x^2$
2. Then $du = 2x\,dx$
3. Rewrite the integral:

$$\int e^u\,du$$

4. Integrate:

$$e^u + C$$

5. Substitute back:

$$e^{x^2} + C$$

That’s it. Clean and direct.

On a no‑calculator section, this is very common. The work needs to show the substitution clearly to earn full credit on an FRQ.

Definite Integrals

Now consider:

$$\int_{0}^{1} 4x (2x^2 + 3)^5 dx$$

Let $u = 2x^2 + 3$, so $du = 4x\,dx$.

Changing the Limits (preferred)

When $x = 0$:
$u = 3$

When $x = 1$:
$u = 5$

Rewrite:

$$\int_{3}^{5} u^5\,du$$

Integrate:

$$\frac{u^6}{6} \Big|_{3}^{5}$$

No back‑substitution needed.

This method avoids algebra mistakes at the end. The AP graders love when everything stays in terms of $u$.

4. Rearranging Before Substituting

Sometimes it doesn’t look ready.

You might need to:

• Rewrite radicals as powers
  $\sqrt{g(x)} = (g(x))^{1/2}$
• Pull out constants
• Split fractions
• Rewrite denominators using negative exponents

Example:

$$\int \frac{5x}{(x^2+1)^3} dx$$

Let $u = x^2 + 1$, so $du = 2x\,dx$.

You only have $5x\,dx$, not $2x\,dx$, so adjust:

$$= \frac{5}{2} \int \frac{2x}{(x^2+1)^3} dx$$

Now substitution works cleanly.

The algebra step is often where students lose points, not the substitution itself.

5. Common Mistakes and AP Traps

• Forgetting to change limits after switching to $u$
• Mixing $x$ and $u$ in the same integral
• Forgetting the constant $+C$
• Ignoring absolute value in log results

If you see:

$$\int \frac{g'(x)}{g(x)} dx$$

the answer is:

$$\ln |g(x)| + C$$

That absolute value matters.

Also, if substitution makes the integral messier, you chose the wrong $u$. Back up and rethink the inside function.