Calculus 2 Complete Practice Exam

1

Integration by Parts

\(\int u\,dv = uv - \int v\,du\)

🧠 LIATE Rule (choose u in this order)

Logarithmic → Inverse trig → Algebraic → Trigonometric → Exponential

✏️ Example

Evaluate \(\int x e^x\,dx\)
Let \(u = x\), \(dv = e^x dx\) → \(du = dx\), \(v = e^x\)

Answer: \(xe^x - e^x + C = e^x(x-1) + C\)

2

Trigonometric Integrals

\(\sin^2\theta = \frac{1-\cos 2\theta}{2}\) \(\cos^2\theta = \frac{1+\cos 2\theta}{2}\)
\(\int \tan x\,dx = \ln|\sec x| + C\) \(\int \sec x\,dx = \ln|\sec x + \tan x| + C\)

🧠 Strategy for \(\int \sin^m x \cos^n x\,dx\)

• Odd power of sin: save one sin, convert rest with \(\cos^2 = 1 - \sin^2\)
• Odd power of cos: save one cos, convert rest with \(\sin^2 = 1 - \cos^2\)
• Both even: use half-angle identities

✏️ Example

\(\int \cos^3 x\,dx\) = \(\int \cos^2 x \cdot \cos x\,dx\) = \(\int (1-\sin^2 x)\cos x\,dx\)

Answer: \(\sin x - \frac{\sin^3 x}{3} + C\)

3

Trigonometric Substitution

\(\sqrt{a^2 - x^2}\): let \(x = a\sin\theta\)
\(\sqrt{a^2 + x^2}\): let \(x = a\tan\theta\)
\(\sqrt{x^2 - a^2}\): let \(x = a\sec\theta\)

✏️ Example

\(\int \frac{dx}{\sqrt{4-x^2}}\): let \(x = 2\sin\theta\), \(dx = 2\cos\theta\,d\theta\)

Answer: \(\arcsin\!\left(\frac{x}{2}\right) + C\)

4

Partial Fractions

\(\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}\)
Repeated: \(\frac{P(x)}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}\)
Irreducible quadratic: \(\frac{P(x)}{x^2+bx+c} = \frac{Ax+B}{x^2+bx+c}\)

✏️ Example

\(\int \frac{1}{x^2-1}\,dx = \int\!\left(\frac{1/2}{x-1} - \frac{1/2}{x+1}\right)dx\)

Answer: \(\frac{1}{2}\ln\!\left|\frac{x-1}{x+1}\right| + C\)

5

Improper Integrals

\(\int_a^{\infty} f(x)\,dx = \lim_{t\to\infty}\int_a^t f(x)\,dx\)

\(p\)-Test: \(\int_1^{\infty}\frac{1}{x^p}dx\) converges iff \(p > 1\)

✏️ Example

\(\int_1^{\infty} \frac{1}{x^2}\,dx = \lim_{t\to\infty}\left[-\frac{1}{x}\right]_1^t = 0 - (-1) = 1\)

Answer: 1 (converges)

6

Sequences and Series

Geometric series: \(\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}\), \(|r|<1\)

Divergence Test: if \(\lim_{n\to\infty} a_n \neq 0\), series diverges

\(p\)-Series: \(\sum \frac{1}{n^p}\) converges iff \(p > 1\)

🧠 Convergence Tests Summary

Ratio Test → Exponentials/Factorials
Root Test → \(n\)-th powers
Integral Test → positive, decreasing, continuous
Comparison/Limit Comparison → algebraic terms
Alternating Series Test → alternating signs

7

Power Series and Taylor Series

Radius of convergence: \(R = \lim_{n\to\infty}\left|\frac{a_n}{a_{n+1}}\right|\)

\(e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}\) \(\sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}\)

\(\cos x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}\) \(\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n\)

✏️ Example

First three nonzero terms of \(e^x\):

\(1 + x + \frac{x^2}{2!} + \cdots\)

8

Polar Coordinates and Parametric Curves

Area in polar: \(A = \frac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta\)

Arc length (parametric): \(L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt\)

Arc length (polar): \(L = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta\)

9

Applications of Integration

Arc length: \(L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx\)

Volume (disk): \(V = \pi\int_a^b [f(x)]^2\,dx\)

Volume (shell): \(V = 2\pi\int_a^b x\,f(x)\,dx\)

Surface area: \(S = 2\pi\int_a^b f(x)\sqrt{1+[f'(x)]^2}\,dx\)

10

First-Order Differential Equations

Separable: \(\frac{dy}{dx} = g(x)h(y)\) → \(\int\frac{dy}{h(y)} = \int g(x)\,dx\)

Linear: \(y' + P(x)y = Q(x)\), integrating factor \(\mu = e^{\int P\,dx}\)

✏️ Example

\(\frac{dy}{dx} = xy\) → \(\frac{dy}{y} = x\,dx\) → \(\ln|y| = \frac{x^2}{2} + C\)

Answer: \(y = Ae^{x^2/2}\)

Calculus 2

📚 Concept Review & Key Formulas