1) Introduction: A New Tool for Integration 📚
So far, we’ve used basic rules to find indefinite integrals, but some integrals look tricky or impossible with those alone. That’s where u-substitution comes in! It’s a technique that simplifies complex integrals by changing variables, making them easier to solve. Think of it as a magic wand for integration!
In this topic, we’ll learn:
- What u-Substitution Is: How to pick a new variable to simplify the integral.
- Step-by-Step Process: A clear method to apply substitution.
- Examples: Putting it into action with different functions.
Quick Recap: An indefinite integral \( \int f(x) \, dx = F(x) + C \) requires finding \( F'(x) = f(x) \).
2) What Is u-Substitution? 🎓
U-substitution is like a transformation that rewrites an integral in terms of a new variable \( u \). We choose \( u \) to be a part of the integrand that, when differentiated, helps simplify the problem. The goal is to turn a complicated integral into one we already know how to solve.
Definition 10.1: u-Substitution
To integrate \( \int f(g(x)) g'(x) \, dx \), set \( u = g(x) \). Then \( du = g'(x) \, dx \), and the integral becomes \( \int f(u) \, du \), which we can solve using basic rules.
The key is to express \( dx \) in terms of \( du \) and adjust the integral accordingly. Let’s see how it works!
Example 1: Basic u-Substitution
Find \( \int 2x e^{x^2} \, dx \).
Let’s try \( u = x^2 \). Then \( du = 2x \, dx \), which matches the integrand!
- Rewrite: \( \int e^{x^2} \cdot 2x \, dx = \int e^u \, du \).
- Integrate: \( \int e^u \, du = e^u + C \).
- Substitute back: \( e^u = e^{x^2} \), so the answer is \( e^{x^2} + C \).
Answer: \( \int 2x e^{x^2} \, dx = e^{x^2} + C \).
Example 2: Adjusting the Expression
Find \( \int x^2 e^{x^3} \, dx \).
Let \( u = x^3 \). Then \( du = 3x^2 \, dx \), so \( x^2 \, dx = \frac{du}{3} \).
- Rewrite: \( \int e^{x^3} \cdot x^2 \, dx = \int e^u \cdot \frac{du}{3} = \frac{1}{3} \int e^u \, du \).
- Integrate: \( \frac{1}{3} e^u + C \).
- Substitute back: \( \frac{1}{3} e^{x^3} + C \).
Answer: \( \int x^2 e^{x^3} \, dx = \frac{1}{3} e^{x^3} + C \).
3) Step-by-Step Process for u-Substitution 📐
Let’s break u-substitution into clear steps to make it foolproof:
Procedure for u-Substitution
- Choose \( u \): Pick a part of the integrand that, when differentiated, appears elsewhere (e.g., inside an exponent or a product).
- Compute \( du \): Differentiate \( u \) with respect to \( x \) to find \( du = \frac{du}{dx} \, dx \).
- Rewrite the Integral: Substitute \( u \) and \( du \) into the integral, adjusting coefficients if needed.
- Integrate: Solve the new integral in terms of \( u \).
- Substitute Back: Replace \( u \) with the original expression in \( x \).
Example 3: Following the Steps
Find \( \int (x + 1) e^{x^2 + 2x} \, dx \).
- Choose \( u \): Let \( u = x^2 + 2x \).
- Compute \( du \): \( du = (2x + 2) \, dx \), so \( (x + 1) \, dx = \frac{du}{2} \) (since \( 2x + 2 = 2(x + 1) \)).
- Rewrite: \( \int e^{x^2 + 2x} \cdot (x + 1) \, dx = \int e^u \cdot \frac{du}{2} = \frac{1}{2} \int e^u \, du \).
- Integrate: \( \frac{1}{2} e^u + C \).
- Substitute Back: \( \frac{1}{2} e^{x^2 + 2x} + C \).
Answer: \( \int (x + 1) e^{x^2 + 2x} \, dx = \frac{1}{2} e^{x^2 + 2x} + C \).
Example 4: Trigonometric Substitution
Find \( \int \sin(3x) \, dx \).
- Choose \( u \): Let \( u = 3x \).
- Compute \( du \): \( du = 3 \, dx \), so \( dx = \frac{du}{3} \).
- Rewrite: \( \int \sin(3x) \, dx = \int \sin(u) \cdot \frac{du}{3} = \frac{1}{3} \int \sin(u) \, du \).
- Integrate: \( \frac{1}{3} (-\cos(u)) + C \).
- Substitute Back: \( \frac{1}{3} (-\cos(3x)) + C = -\frac{1}{3} \cos(3x) + C \).
Answer: \( \int \sin(3x) \, dx = -\frac{1}{3} \cos(3x) + C \).
4) Advanced Examples 🔍
Example 5: Nested Function
Find \( \int x^2 \cos(x^3) \, dx \).
- Choose \( u \): Let \( u = x^3 \).
- Compute \( du \): \( du = 3x^2 \, dx \), so \( x^2 \, dx = \frac{du}{3} \).
- Rewrite: \( \int \cos(x^3) \cdot x^2 \, dx = \int \cos(u) \cdot \frac{du}{3} = \frac{1}{3} \int \cos(u) \, du \).
- Integrate: \( \frac{1}{3} \sin(u) + C \).
- Substitute Back: \( \frac{1}{3} \sin(x^3) + C \).
Answer: \( \int x^2 \cos(x^3) \, dx = \frac{1}{3} \sin(x^3) + C \).
Example 6: Multiple Adjustments
Find \( \int (2x + 3) e^{x^2 + 3x} \, dx \).
- Choose \( u \): Let \( u = x^2 + 3x \).
- Compute \( du \): \( du = (2x + 3) \, dx \), which matches perfectly!
- Rewrite: \( \int e^{x^2 + 3x} \cdot (2x + 3) \, dx = \int e^u \, du \).
- Integrate: \( e^u + C \).
- Substitute Back: \( e^{x^2 + 3x} + C \).
Answer: \( \int (2x + 3) e^{x^2 + 3x} \, dx = e^{x^2 + 3x} + C \).
5) Practice Questions 🎯
Fundamental Practice Questions 🌱
Instructions: Find the general antiderivative (indefinite integral) for each function using u-substitution. 📚
\( \int x e^{x^2} \, dx \)
\( \int \cos(2x) \, dx \)
\( \int x^2 \sin(x^3) \, dx \)
\( \int (x + 1) e^{x^2 + 2x + 1} \, dx \)
\( \int \sin(4x) \, dx \)
\( \int x^3 e^{x^4} \, dx \)
\( \int \cos(3x + 1) \, dx \)
\( \int x \cos(x^2 + 1) \, dx \)
\( \int (2x + 1) e^{x^2 + x} \, dx \)
\( \int x^2 e^{x^3 + 2} \, dx \)
\( \int \sin(5x) \, dx \)
Challenging Practice Questions 🌟
Instructions: These require deeper understanding or advanced application of u-substitution. 🧠
Find \( \int x^2 e^{x^3 - 1} \, dx \) and verify by differentiating.
Compute \( \int (3x^2 + 2x) \cos(x^3 + x^2) \, dx \) using u-substitution.
Determine \( \int \sin^2(x) \cos(x) \, dx \) with a suitable substitution (hint: try \( u = \sin(x) \)).
Evaluate \( \int x (x^2 + 1)^3 \, dx \) and check your result.
Explain how to choose \( u \) for \( \int x^2 \sqrt{x^3 + 1} \, dx \) and find the integral.
6) Summary & Cheat Sheet 📋
6.1) u-Substitution
Set \( u = g(x) \), then \( du = g'(x) \, dx \), and rewrite \( \int f(g(x)) g'(x) \, dx = \int f(u) \, du \).
6.2) Steps
- Choose \( u \) (a part to simplify).
- Find \( du \).
- Rewrite and integrate in terms of \( u \).
- Substitute back to \( x \).
6.3) Tip
Look for a function and its derivative in the integrand to guide your \( u \) choice.
You’ve mastered u-substitution! Next, we’ll explore definite integrals. 🎉