Calculus 2 Cheat Sheet

1. Integration Techniques

1.1 Integration by Parts

Integration by parts is based on the product rule for differentiation. The formula is:

\[ \int u \, dv = uv - \int v \, du \]

Example:

Compute \( \int x \cdot e^x \, dx \) using integration by parts.

Let \(u = x\) and \(dv = e^x \, dx\).
Then \(du = dx\) and \(v = e^x\).
Apply the formula: \( \int x \cdot e^x \, dx = x \cdot e^x - \int e^x \, dx \).
Integrate: \( x \cdot e^x - e^x + C \).
Result: \( \int x \cdot e^x \, dx = e^x(x - 1) + C \).

1.2 Trigonometric Integrals

Trigonometric integrals involve integrals of products of sine, cosine, and other trigonometric functions. Common strategies include using trigonometric identities to simplify the integrand.

Example:

Compute \( \int \sin^2(x) \, dx \).

Use the identity \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \).
Rewrite the integral: \( \int \frac{1 - \cos(2x)}{2} \, dx \).
Integrate: \( \frac{1}{2} \int (1 - \cos(2x)) \, dx = \frac{1}{2} \left( x - \frac{\sin(2x)}{2} \right) + C \).
Result: \( \int \sin^2(x) \, dx = \frac{x}{2} - \frac{\sin(2x)}{4} + C \).

1.3 Trigonometric Substitution

Trigonometric substitution is used to simplify integrals involving square roots of quadratic expressions. The substitution depends on the form of the quadratic expression.

Example:

Compute \( \int \frac{dx}{\sqrt{a^2 - x^2}} \).

Use the substitution \( x = a \sin(\theta) \), \( dx = a \cos(\theta) \, d\theta \).
The integral becomes \( \int \frac{a \cos(\theta) \, d\theta}{\sqrt{a^2 - a^2 \sin^2(\theta)}} = \int d\theta \).
Integrate: \( \theta + C \).
Substitute back: \( \theta = \arcsin\left(\frac{x}{a}\right) \).
Result: \( \int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\left(\frac{x}{a}\right) + C \).

1.4 Partial Fraction Decomposition

Partial fraction decomposition is used to integrate rational functions by expressing them as a sum of simpler fractions.

Example:

Compute \( \int \frac{2x + 3}{x^2 - x - 6} \, dx \) using partial fractions.

Factor the denominator: \( x^2 - x - 6 = (x - 3)(x + 2) \).
Decompose into partial fractions: \( \frac{2x + 3}{(x - 3)(x + 2)} = \frac{A}{x - 3} + \frac{B}{x + 2} \).
Solve for \(A\) and \(B\): \( 2x + 3 = A(x + 2) + B(x - 3) \).
Substitute back and integrate each term.
Result: \( \int \frac{2x + 3}{x^2 - x - 6} \, dx = \ln|x - 3| - \ln|x + 2| + C \).

2. Sequences and Series

2.1 Convergence of Sequences

A sequence \(\{a_n\}\) converges to a limit \(L\) if, for every \(\epsilon > 0\), there exists an integer \(N\) such that for all \(n > N\), \(|a_n - L| < \epsilon\).

Example:

Determine if the sequence \(a_n = \frac{1}{n}\) converges.

As \(n\) increases, \(a_n = \frac{1}{n}\) gets closer to 0.
For any \(\epsilon > 0\), choose \(N = \frac{1}{\epsilon}\).
If \(n > N\), then \(\frac{1}{n} < \epsilon\), so the sequence converges to 0.
Result: The sequence converges to 0.

2.2 Series and Convergence Tests

A series is the sum of the terms of a sequence: \(\sum_{n=1}^{\infty} a_n\). Common convergence tests include:

Geometric Series Test: Converges if \(|r| < 1\).
p-Series Test: \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) converges if \(p > 1\).
Comparison Test: Compare with a known convergent or divergent series.
Ratio Test: Converges if \(\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1\).

Example:

Determine if the series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges.

This is a p-series with \(p = 2\).
Since \(p > 1\), the series converges by the p-Series Test.
Result: The series converges.

2.3 Power Series

A power series is an infinite series of the form \( \sum_{n=0}^{\infty} c_n (x - a)^n \), where \(c_n\) are constants. The series converges within a radius of convergence \(R\), which can be found using the Ratio Test.

Example:

Find the radius of convergence for the series \( \sum_{n=1}^{\infty} \frac{x^n}{n} \).

Apply the Ratio Test: \( \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \left|\frac{x^{n+1}}{n+1} \cdot \frac{n}{x^n}\right| = \left|\frac{x}{1}\right| \).
For convergence, \( \left|\frac{x}{1}\right| < 1 \), so \( |x| < 1 \).
Result: The radius of convergence is \(R = 1\).

2.4 Taylor and Maclaurin Series

A Taylor series is a power series representation of a function centered at \(x = a\):

\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n \]

A Maclaurin series is a Taylor series centered at \(x = 0\).

Example:

Find the Maclaurin series for \( e^x \).

For \( e^x \), all derivatives are \( e^x \), so \( f^{(n)}(0) = 1 \) for all \(n\).
The Maclaurin series is: \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \).
Result: \( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \).

3. Parametric Equations and Polar Coordinates

3.1 Parametric Equations

Parametric equations represent curves by expressing the coordinates \(x\) and \(y\) as functions of a parameter \(t\). The slope of the curve is given by:

\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \]

Example:

Find \( \frac{dy}{dx} \) for the curve given by \( x(t) = t^2 \) and \( y(t) = t^3 \).

Compute \( \frac{dx}{dt} = 2t \) and \( \frac{dy}{dt} = 3t^2 \).
Then \( \frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2} \).
Result: \( \frac{dy}{dx} = \frac{3t}{2} \).

3.2 Polar Coordinates

In polar coordinates, a point in the plane is represented by \( (r, \theta) \), where \(r\) is the distance from the origin and \( \theta \) is the angle from the positive x-axis. The relationships between Cartesian and polar coordinates are:

\[ x = r \cos(\theta), \quad y = r \sin(\theta) \]

The area under a curve in polar coordinates is given by:

\[ A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta \]

Example:

Find the area enclosed by the curve \( r(\theta) = 2 + 2\sin(\theta) \).

Set up the integral: \( A = \frac{1}{2} \int_{0}^{2\pi} (2 + 2\sin(\theta))^2 \, d\theta \).
Expand and integrate: \( \int_{0}^{2\pi} (4 + 8\sin(\theta) + 4\sin^2(\theta)) \, d\theta \).
Use trigonometric identities to simplify and integrate.
Result: The area is \( 8\pi \) square units.

4. Differential Equations

4.1 First-Order Differential Equations

A first-order differential equation is an equation involving the first derivative of an unknown function. It can often be solved by separation of variables or using an integrating factor.

Example:

Solve the differential equation \( \frac{dy}{dx} = xy \) using separation of variables.

Separate variables: \( \frac{dy}{y} = x \, dx \).
Integrate both sides: \( \ln|y| = \frac{x^2}{2} + C \).
Exponentiate to solve for \(y\): \( y = e^{\frac{x^2}{2} + C} = Ce^{\frac{x^2}{2}} \).
Result: \( y = Ce^{\frac{x^2}{2}} \), where \(C\) is the constant of integration.

4.2 Second-Order Differential Equations

Second-order differential equations involve the second derivative of the unknown function. A common method of solution is the characteristic equation, especially for linear equations with constant coefficients.

Example:

Solve the differential equation \( y'' - 4y' + 4y = 0 \).

Write the characteristic equation: \( r^2 - 4r + 4 = 0 \).
Factor: \( (r - 2)^2 = 0 \), so \(r = 2\) (a repeated root).
The general solution is \( y = (C_1 + C_2x)e^{2x} \).
Result: \( y = (C_1 + C_2x)e^{2x} \), where \(C_1\) and \(C_2\) are constants.