Mastering Advanced Integration: Substitution and Parts

Step 2: Differentiate $u$ to find $du$: $\frac{du}{dx} = 2x \implies du = 2x \, dx$

Step 3: Substitute $u$ and $du$ into the integral: $\int e^u \, du$

Step 4: Integrate: $e^u + C$

Step 5: Back-substitute $u = x^2$: $e^{x^2} + C$

Exam Tip: If your substitution leaves behind an $x$ that cannot be easily converted to $u$, you may have chosen the wrong $u$, or you may need to solve your $u$-equation for $x$ to perform a "change of variables".

Integration by Parts: The Reverse Product Rule

Sometimes, an integrand consists of two different types of functions multiplied together, such as an algebraic function times a logarithmic function (e.g., $x \ln(x)$). In these cases, $u$-substitution usually fails, and we turn to Integration by Parts.

This technique is derived directly from the Product Rule for differentiation. The formula is: $\int u \, dv = uv - \int v \, du$

The LIATE Rule

The most difficult part of Integration by Parts is deciding which part of your integrand should be $u$ and which part should be $dv$. If you pick the wrong $u$, the resulting integral $\int v \, du$ might become even more complicated than the original.

To make this decision, mathematicians use the LIATE mnemonic. You should choose $u$ based on which function appears first in this list:

1. Logarithmic functions (e.g., $\ln(x)$, $\log_a(x)$)
2. Inverse trigonometric functions (e.g., $\arctan(x)$, $\arcsin(x)$)
3. Algebraic functions (e.g., $x^2$, $3x$, $x^{1/2}$)
4. Trigonometric functions (e.g., $\sin(x)$, $\cos(x)$)
5. Exponential functions (e.g., $e^x$, $2^x$)

By following LIATE, you prioritize functions that become simpler when differentiated (like logs and inverse trig) as your $u$, and functions that are easy to integrate (like exponentials and trig) as your $dv$.

Worked Example: Integration by Parts

Evaluate the following integral: $\int x \cos(x) \, dx$

Step 1: Choose $u$ and $dv$ using LIATE.

• $x$ is an Algebraic function.
• $\cos(x)$ is a Trigonometric function. Since A comes before T in LIATE, we set: $u = x$ $dv = \cos(x) \, dx$

Step 2: Differentiate $u$ and integrate $dv$. $\frac{du}{dx} = 1 \implies du = dx$ $\int dv = \int \cos(x) \, dx \implies v = \sin(x)$

Step 3: Apply the formula $\int u \, dv = uv - \int v \, du$. $\int x \cos(x) \, dx = (x)(\sin(x)) - \int \sin(x) \, dx$

Step 4: Evaluate the remaining integral. $\int x \cos(x) \, dx = x \sin(x) - (-\cos(x)) + C$ $\int x \cos(x) \, dx = x \sin(x) + \cos(x) + C$

Summary Comparison

Mastering these two techniques provides the foundation for solving nearly all standard integral problems encountered in calculus. Practice identifying the "structure" of the integrand before you start writing—ask yourself: "Is this a composition (Substitution) or a product (Parts)?".