🔗 U-Substitution Method

What is U-Substitution?

U-substitution is the reverse of the chain rule for derivatives. It's the most important integration technique you'll learn!

💡 Key Idea: If an integral has the form "function times its derivative," use u-substitution!

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The Chain Rule (Review)

Remember the chain rule for derivatives:

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

Example: $\frac{d}{dx}[(x^2+1)^5] = 5(x^2+1)^4 \cdot 2x$

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Reversing the Chain Rule

If we integrate both sides:

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

This is the foundation of u-substitution!

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The U-Substitution Process

Step-by-Step Method

Step 1: Identify $u$ (the "inside function")

• Look for a composite function
• Choose $u = g(x)$

Step 2: Find $du$

• Calculate $du = g'(x)\,dx$

Step 3: Rewrite the integral in terms of $u$

• Replace all $x$ terms with $u$ terms
• Replace $dx$ with $du$

Step 4: Integrate with respect to $u$

Step 5: Substitute back to $x$

• Replace $u$ with the original expression

Step 6: Add $+C$

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Example 1: Basic U-Substitution

Evaluate $\int 2x(x^2+1)^5\,dx$

Step 1: Choose $u$

Let $u = x^2 + 1$ (the inside function)

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Step 2: Find $du$

$\frac{du}{dx} = 2x$

$du = 2x\,dx$

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Step 3: Rewrite the integral

Notice that we have $2x\,dx$ in the original integral!

$\int 2x(x^2+1)^5\,dx = \int (x^2+1)^5 \cdot 2x\,dx$

$= \int u^5\,du$

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Step 4: Integrate

$= \frac{u^6}{6} + C$

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Step 5: Substitute back

$= \frac{(x^2+1)^6}{6} + C$

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Check (using chain rule): $\frac{d}{dx}\left[\frac{(x^2+1)^6}{6}\right] = \frac{6(x^2+1)^5 \cdot 2x}{6} = 2x(x^2+1)^5$ ✓

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Example 2: When You Need to Adjust

Evaluate $\int x(x^2+1)^5\,dx$

Step 1: Choose $u$

$u = x^2 + 1$

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Step 2: Find $du$

$du = 2x\,dx$

So: $x\,dx = \frac{1}{2}du$

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Step 3: Rewrite

$\int x(x^2+1)^5\,dx = \int (x^2+1)^5 \cdot x\,dx$

$= \int u^5 \cdot \frac{1}{2}du$

$= \frac{1}{2}\int u^5\,du$

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Step 4: Integrate

$= \frac{1}{2} \cdot \frac{u^6}{6} + C = \frac{u^6}{12} + C$

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Step 5: Substitute back

$= \frac{(x^2+1)^6}{12} + C$

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Pattern Recognition

Type 1: Exact Match

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

The derivative $g'(x)$ appears exactly!

Example: $\int 2x(x^2+1)^5\,dx$

• $u = x^2+1$, $du = 2x\,dx$ ✓

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Type 2: Constant Multiple Off

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

The derivative is off by a constant factor.

Example: $\int x(x^2+1)^5\,dx$

• $u = x^2+1$, $du = 2x\,dx$
• Have $x\,dx = \frac{1}{2}du$ ✓

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Type 3: Won't Work

If after substitution you still have $x$ terms mixed with $u$, u-substitution won't work (try different $u$ or different method).

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Trigonometric Examples

Example 3: Sine and Cosine

Evaluate $\int \sin x \cos^4 x\,dx$

Step 1: Choose $u$

$u = \cos x$ (we have its derivative $\sin x$ in the integral!)

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Step 2: Find $du$

$\frac{du}{dx} = -\sin x$

$du = -\sin x\,dx$

$\sin x\,dx = -du$

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Step 3: Rewrite

$\int \sin x \cos^4 x\,dx = \int \cos^4 x \cdot \sin x\,dx$

$= \int u^4 \cdot (-du) = -\int u^4\,du$

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Step 4: Integrate

$= -\frac{u^5}{5} + C$

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Step 5: Substitute back

$= -\frac{\cos^5 x}{5} + C$

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Exponential Examples

Example 4: Exponential with Chain Rule

Evaluate $\int xe^{x^2}\,dx$

Step 1: Choose $u$

$u = x^2$ (the exponent)

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Step 2: Find $du$

$du = 2x\,dx$

$x\,dx = \frac{1}{2}du$

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Step 3: Rewrite

$\int xe^{x^2}\,dx = \int e^{x^2} \cdot x\,dx$

$= \int e^u \cdot \frac{1}{2}du = \frac{1}{2}\int e^u\,du$

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Step 4: Integrate

$= \frac{1}{2}e^u + C$

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Step 5: Substitute back

$= \frac{1}{2}e^{x^2} + C$

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Example 5: Requiring Algebraic Manipulation

Evaluate $\int \frac{x}{x^2+1}\,dx$

Step 1: Choose $u$

$u = x^2 + 1$ (the denominator)

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Step 2: Find $du$

$du = 2x\,dx$

$x\,dx = \frac{1}{2}du$

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Step 3: Rewrite

$\int \frac{x}{x^2+1}\,dx = \int \frac{1}{x^2+1} \cdot x\,dx$

$= \int \frac{1}{u} \cdot \frac{1}{2}du = \frac{1}{2}\int \frac{1}{u}\,du$

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Step 4: Integrate

$= \frac{1}{2}\ln|u| + C$

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Step 5: Substitute back

$= \frac{1}{2}\ln|x^2+1| + C$

Since $x^2+1 > 0$ always, we can write:

$= \frac{1}{2}\ln(x^2+1) + C$

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When to Use U-Substitution

Good Candidates

Look for these patterns:

1. Composite function with its derivative present

   • $\int (x^2+1)^5 \cdot 2x\,dx$

2. Fraction where numerator is derivative of denominator

   • $\int \frac{2x}{x^2+1}\,dx$

3. Trig functions with derivatives

   • $\int \tan x\,dx = \int \frac{\sin x}{\cos x}\,dx$ (let $u = \cos x$)

4. Exponentials with derivatives of exponent

   • $\int xe^{x^2}\,dx$

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Common U-Choices

| Integral Type | Common Choice for $u$ | |---------------|----------------------| | $(ax+b)^n$ | $u = ax+b$ | | $f'(x) \cdot [f(x)]^n$ | $u = f(x)$ | | $\frac{f'(x)}{f(x)}$ | $u = f(x)$ (gives $\ln$ ) | | $e^{f(x)} \cdot f'(x)$ | $u = f(x)$ | | $\sin^n x \cos x$ | $u = \sin x$ | | $\cos^n x \sin x$ | $u = \cos x$ |

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⚠️ Common Mistakes

Mistake 1: Choosing Wrong $u$

WRONG: For $\int x(x^2+1)^5\,dx$, choosing $u = x$

RIGHT: Choose $u = x^2+1$ (the composite function)

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Mistake 2: Forgetting to Adjust Constants

WRONG: $u = x^2+1$, $du = 2x\,dx$, and writing $\int u^5\,du$ for $\int x(x^2+1)^5\,dx$

RIGHT: Need $\frac{1}{2}$ factor! $\int u^5 \cdot \frac{1}{2}du$

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Mistake 3: Not Substituting Everything

After substitution, you should have only $u$ and $du$ - no $x$ remaining!

If $x$ remains, choose different $u$ or method won't work.

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Mistake 4: Forgetting to Substitute Back

WRONG: Final answer $\frac{u^6}{12} + C$

RIGHT: Final answer $\frac{(x^2+1)^6}{12} + C$

Always convert back to the original variable!

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Mistake 5: Wrong Sign

When $du = -\sin x\,dx$, we have $\sin x\,dx = -du$ (negative!)

Don't forget the negative sign!

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U-Substitution Checklist

✓ Choose $u$ = inner function or complicated part

✓ Find $du$ and solve for needed differential

✓ Rewrite integral completely in terms of $u$

✓ Check: No $x$ should remain after substitution

✓ Integrate using basic formulas

✓ Substitute back to original variable

✓ Add $+C$

✓ Check by differentiating (if time permits)

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Practice Tips

Tip 1: Look for the Derivative

Ask: "What function's derivative do I see here?"

Tip 2: Try the "Inside" First

For composite functions like $(x^2+1)^5$, try $u =$ inside function first.

Tip 3: Don't Force It

If after substitution you still have messy $x$ and $u$ mixed, try different $u$ or different method.

Tip 4: Practice Pattern Recognition

The more you practice, the faster you'll recognize which $u$ to choose!

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Advanced Example

Evaluate $\int \frac{\ln x}{x}\,dx$

Solution:

Let $u = \ln x$

$du = \frac{1}{x}dx$

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

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📝 Practice Strategy

1. Identify the composite function or complicated part
2. Look for its derivative somewhere in the integral
3. Set $u$ equal to the inner function
4. Calculate $du$ and solve for needed form
5. Substitute completely - no $x$ should remain!
6. Integrate the simpler $u$ integral
7. Substitute back to $x$
8. Always add $+C$
9. Check your answer by differentiating