MATH 101 - 102: Lecture 7 Study Notes

MATH 101 - 102: Lecture 7

Instructor Information

• Instructor: Nima Eslami

• The University of British Columbia, Okanagan Campus

• Department of Mathematics, Computer Science, Physics, and Statistics

• Irving K. Barber Faculty of Science

• Date of Lecture: January 27, 2026

Agenda

• Today we will cover the following integration techniques:

  • u-substitution

  • reverse u-substitution

  • rationalizing substitutions

  • integration by parts

First Technique of Integration: U-Substitution

The U-Substitution Strategy for Definite Integrals

To evaluate an integral of the form $\int_{a}^{b} f(g(x)) g'(x) \, dx$, follow these steps:

1. Choose the substitution:

   • Set $u = g(x)$.

2. Differentiate:

   • Differentiate both sides to relate $du$ and $dx$:
     $du = g'(x) \, dx$ (equivalently, $dx = \frac{1}{g'(x)} \, du$).

3. Rewrite the integrand:

   • Substitute $u = g(x)$ and $du = g'(x) \, dx$ into the integral so that everything is expressed in terms of $u$:
     $\int_{a}^{b} f(g(x)) g'(x) \, dx = \int f(u) \, du$.

4. Change the bounds:

   • Convert the limits of integration from x-bounds to u-bounds using the substitution:

   • When $x = a$, then $u = g(a)$;

   • When $x = b$, then $u = g(b)$.

   • Therefore,
     $\int<em>{a}^{b} f(g(x)) g'(x) \, dx = \int</em>{g(a)}^{g(b)} f(u) \, du$.

5. Integrate and evaluate:

   • If $F'(u) = f(u)$, then
     $\int<em>{g(a)}^{g(b)} f(u) \, du = F(u) \bigg|</em>{g(a)}^{g(b)} = F(g(b)) - F(g(a))$.

Note: Substituting Back to the Original Variable

• If you change the bounds to u-bounds (recommended), typically you do not substitute back to $x$ inside the antiderivative; simply evaluate using $u = g(a)$ and $u = g(b)$.

• If you keep the original bounds $a$ and $b$ in $x$, you must substitute back $u = g(x)$ before evaluating:
  $\int<em>{a}^{b} f(g(x)) g'(x) \, dx = F(g(x)) \bigg|</em>{a}^{b} = F(g(b)) - F(g(a))$.

Indefinite Integrals and U-Substitution

• For indefinite integrals, there are no bounds to change. After substituting $u = g(x)$ and $du = g'(x) \, dx$, follow these steps:

  1. Integrate with respect to $u$:
     $\int f(g(x)) g'(x) \, dx = \int f(u) \, du$.

  2. After integration, substitute back:
     $= F(u) + C$.

  3. Replace $u$ with $g(x)$ including the constant of integration:
     $= F(g(x)) + C$.

Example

1. Calculate the following indefinite integral:

   • $\int 2x \sin(x^2 + 5) \, dx$.

Additional Examples

2. Calculate the following indefinite integral:

   • $\int \frac{8 \sin(x)}{\cos^2(2x) + 1} \, dx$.

3. Calculate the following definite integral:

   • $\int_{0}^{10} (2x + 9)(2x^2 + 9x) \, dx$.

4. Calculate the following definite integral:

   • $\int_{3}^{L} xe^{t^2} \, dx$.

5. Calculate the following definite integral:

   • $\int_{0}^{T} (3 + 4x) \, dx$ from $x^2 + 1$.

Practice Questions

• Calculate the following definite integrals:

  1. $\int_{0}^{\frac{\pi}{4}} 4 \sec^2(\theta)(\tan^7(\theta) + 1) \, d\theta$.

  2. $\int_{-2}^{1} \frac{w + 5}{2w^2 + 1} \, dw$.

  3. $\int_{e}^{e^2} \left(5 + \ln(2x) \right) \left(4 + (\ln(2x))^3 \right)7x \, dx$.

  4. $\int_{1}^{4} (3 - \sqrt{x}) 2\sqrt{x} \, dx$.

A Technical Note

• Suppose that $F(x)$ is an antiderivative of $f(x)$, and consider $\int f(ax + b) \, dx$, where $a, b \in \mathbb{R}$ and $a \neq 0$.

• Make the change of variables:

  • Set $u = ax + b$.

  • Thus, $du = a \, dx$, leading to $dx = \frac{1}{a} \, du$.

• Now substitute and integrate:
  $\int f(ax + b) \, dx = \int f(u) \cdot \frac{1}{a} \, du$.

• Therefore, it results in:
  $= \frac{F(u)}{a} + C = \frac{F(ax + b)}{a} + C$.

• Example: Evaluate $\int \sin(3x + 2) \, dx$.

Reverse U-Substitution

Example

• Calculate the integral:
  $\int \frac{x}{1+x} \, dx$.

Additional Examples

1. Calculate the following indefinite integral:

   • $\int \sqrt{2 - \sqrt{x}} \, dx$.

2. Calculate the following definite integral:

   • $\int_{0}^{2} \frac{3x-2}{2x + 1} \, dx$.

Practice Questions

• Calculate the following integrals using reverse u-substitution:

  1. $\int (2 \sqrt{3} x^3 + \sqrt{3} x) \, dx$.

  2. $\int 5m^2 \sqrt{4 - 2m} \, dm$.

  3. $\int_{2}^{4} \frac{\sqrt{z+2}}{2z-1} \, dz$.

Rationalizing Substitutions

Example

• Use a rationalizing substitution to calculate the following integral:

  • $\int \frac{\sqrt{x}-9}{x} \, dx$.

Additional Examples

• Use a rationalizing substitution to calculate the following integrals:

  1. $\int \frac{1}{x + x^{1/3}} \, dx$.

Integration by Parts

Example

• Compute the integral:

  • $\int x \cos(x) \, dx$.