🔀 Integration by Parts

What is Integration by Parts?

Integration by parts is the reverse of the product rule for derivatives. It's used to integrate products of functions.

💡 Key Idea: Trade one integral for another (hopefully easier) integral!

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The Product Rule (Review)

The product rule for derivatives states:

$\frac{d}{dx}[u(x) \cdot v(x)] = u'(x)v(x) + u(x)v'(x)$

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Deriving the Formula

Integrate both sides of the product rule:

$\int \frac{d}{dx}[uv]\,dx = \int u'v\,dx + \int uv'\,dx$

$uv = \int u'v\,dx + \int uv'\,dx$

Rearranging:

$\int uv'\,dx = uv - \int u'v\,dx$

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The Integration by Parts Formula

$\int u\,dv = uv - \int v\,du$

Or equivalently:

$\int u(x)v'(x)\,dx = u(x)v(x) - \int u'(x)v(x)\,dx$

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How to Apply Integration by Parts

Step-by-Step Process

Step 1: Identify $u$ and $dv$ from the integrand

• Choose what to call $u$
• The rest becomes $dv$

Step 2: Differentiate $u$ to get $du$

Step 3: Integrate $dv$ to get $v$

Step 4: Apply the formula: $\int u\,dv = uv - \int v\,du$

Step 5: Evaluate the new integral $\int v\,du$

Step 6: Simplify and add $+C$

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Choosing $u$ and $dv$

The LIATE Rule (or ILATE)

Choose $u$ in this priority order:

L - Logarithmic functions ($\ln x$, $\log x$)

I - Inverse trig functions ($\arctan x$, $\arcsin x$)

A - Algebraic functions ($x$, $x^2$, $\sqrt{x}$)

T - Trigonometric functions ($\sin x$, $\cos x$, $\tan x$)

E - Exponential functions ($e^x$, $a^x$)

Pick the one that comes first for $u$, the rest becomes $dv$!

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Example 1: Basic Application

Evaluate $\int x e^x\,dx$

Step 1: Choose $u$ and $dv$

Using LIATE:

• Algebraic ($x$) comes before Exponential ($e^x$)

Let $u = x$ (algebraic)

Let $dv = e^x\,dx$ (exponential)

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Step 2: Find $du$

$du = dx$

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Step 3: Find $v$

$v = \int e^x\,dx = e^x$

(Don't add +C to $v$!)

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Step 4: Apply formula

$\int x e^x\,dx = uv - \int v\,du$

$= x \cdot e^x - \int e^x \cdot dx$

$= xe^x - e^x + C$

$= e^x(x-1) + C$

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Check: $\frac{d}{dx}[e^x(x-1)] = e^x(x-1) + e^x(1) = e^x(x-1+1) = xe^x$ ✓

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Example 2: With Logarithm

Evaluate $\int \ln x\,dx$

Trick: Write as $\int 1 \cdot \ln x\,dx$

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Step 1: Choose $u$ and $dv$

Using LIATE:

• Logarithmic ($\ln x$) comes first

Let $u = \ln x$

Let $dv = 1\,dx$ (or just $dx$)

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Step 2: Find $du$

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

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Step 3: Find $v$

$v = \int 1\,dx = x$

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Step 4: Apply formula

$\int \ln x\,dx = uv - \int v\,du$

$= (\ln x)(x) - \int x \cdot \frac{1}{x}\,dx$

$= x\ln x - \int 1\,dx$

$= x\ln x - x + C$

$= x(\ln x - 1) + C$

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Example 3: Trigonometric

Evaluate $\int x\sin x\,dx$

Step 1: Choose $u$ and $dv$

Using LIATE:

• Algebraic ($x$) before Trigonometric ($\sin x$)

Let $u = x$

Let $dv = \sin x\,dx$

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Step 2: Find $du$

$du = dx$

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Step 3: Find $v$

$v = \int \sin x\,dx = -\cos x$

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Step 4: Apply formula

$\int x\sin x\,dx = uv - \int v\,du$

$= x(-\cos x) - \int (-\cos x)\,dx$

$= -x\cos x + \int \cos x\,dx$

$= -x\cos x + \sin x + C$

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Repeated Integration by Parts

Sometimes you need to apply integration by parts multiple times.

Example 4: Apply Twice

Evaluate $\int x^2 e^x\,dx$

First application:

$u = x^2$, $dv = e^x\,dx$

$du = 2x\,dx$, $v = e^x$

$\int x^2 e^x\,dx = x^2 e^x - \int e^x \cdot 2x\,dx$

$= x^2 e^x - 2\int xe^x\,dx$

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Second application on $\int xe^x\,dx$:

$u = x$, $dv = e^x\,dx$

$du = dx$, $v = e^x$

$\int xe^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x$

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Combine:

$\int x^2 e^x\,dx = x^2 e^x - 2(xe^x - e^x) + C$

$= x^2 e^x - 2xe^x + 2e^x + C$

$= e^x(x^2 - 2x + 2) + C$

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Tabular Method (For Repeated Parts)

When you need to apply parts multiple times with polynomial times exponential/trig:

Example: $\int x^3 e^x\,dx$

| Sign | $u$ and derivatives | $dv$ and integrals | |------|-------------------|-------------------| | + | $x^3$ | $e^x$ | | − | $3x^2$ | $e^x$ | | + | $6x$ | $e^x$ | | − | $6$ | $e^x$ | | + | $0$ | $e^x$ |

Answer: $x^3e^x - 3x^2e^x + 6xe^x - 6e^x + C = e^x(x^3 - 3x^2 + 6x - 6) + C$

Multiply diagonally with alternating signs!

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Circular Integration by Parts

Sometimes integration by parts leads back to the original integral!

Example 5: Circular Case

Evaluate $\int e^x \sin x\,dx$

First application:

$u = \sin x$, $dv = e^x\,dx$

$du = \cos x\,dx$, $v = e^x$

$\int e^x \sin x\,dx = e^x \sin x - \int e^x \cos x\,dx$

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Second application on $\int e^x \cos x\,dx$:

$u = \cos x$, $dv = e^x\,dx$

$du = -\sin x\,dx$, $v = e^x$

$\int e^x \cos x\,dx = e^x \cos x - \int e^x(-\sin x)\,dx$

$= e^x \cos x + \int e^x \sin x\,dx$

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Substitute back:

Let $I = \int e^x \sin x\,dx$

$I = e^x \sin x - (e^x \cos x + I)$

$I = e^x \sin x - e^x \cos x - I$

$2I = e^x(\sin x - \cos x)$

$I = \frac{e^x(\sin x - \cos x)}{2} + C$

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When to Use Integration by Parts

Good Candidates

1. Polynomial times exponential: $\int x^n e^x\,dx$
2. Polynomial times trig: $\int x^n \sin x\,dx$ or $\int x^n \cos x\,dx$
3. Logarithm times polynomial: $\int x^n \ln x\,dx$
4. Inverse trig times polynomial: $\int x \arctan x\,dx$
5. Exponential times trig: $\int e^x \sin x\,dx$

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⚠️ Common Mistakes

Mistake 1: Wrong Choice of $u$

WRONG: For $\int x e^x\,dx$, choosing $u = e^x$

This makes things harder! Differentiating $e^x$ doesn't simplify.

RIGHT: Choose $u = x$ (it simplifies when differentiated)

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Mistake 2: Adding +C to $v$

When finding $v = \int dv$, don't add +C yet!

The +C comes at the very end.

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Mistake 3: Forgetting the Minus Sign

The formula is $uv - \int v\,du$ (minus, not plus!)

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Mistake 4: Not Simplifying

After applying the formula, you still need to evaluate $\int v\,du$!

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Mistake 5: Circular Without Solving

If you get the original integral back, don't give up!

Solve algebraically: $I = \ldots - I$ → $2I = \ldots$

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Integration by Parts vs U-Substitution

When to use what?

U-Substitution: Composite function with derivative present

• $\int 2x(x^2+1)^5\,dx$
• $\int xe^{x^2}\,dx$

Integration by Parts: Product of different types of functions

• $\int x e^x\,dx$ (polynomial × exponential)
• $\int x \sin x\,dx$ (polynomial × trig)
• $\int \ln x\,dx$ (logarithm)

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Quick Reference

The Formula

$\int u\,dv = uv - \int v\,du$

LIATE Priority for $u$

L - Logarithmic

I - Inverse trig

A - Algebraic

T - Trigonometric

E - Exponential

Steps

1. Choose $u$ and $dv$ (use LIATE)
2. Find $du$ and $v$
3. Apply formula
4. Evaluate remaining integral
5. Simplify and add +C

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📝 Practice Strategy

1. Use LIATE to choose $u$ (first in list) and $dv$ (the rest)
2. Find $du$ by differentiating $u$
3. Find $v$ by integrating $dv$ (no +C yet!)
4. Write out the formula $uv - \int v\,du$ before substituting
5. Evaluate the new integral $\int v\,du$
6. Check: Does $du$ simplify things? (If not, try different choice)
7. Be ready to apply parts multiple times
8. Watch for circular cases - solve algebraically
9. Add +C at the very end
10. Check by differentiating your answer