Calculus & Analysis Calculator

Basic Integrals

$$\int c \, dx = cx + C$$ (constant rule)

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ for $$n \neq -1$$ (power rule)

$$\int \frac{1}{x} \, dx = \ln|x| + C$$ (logarithmic rule)

$$\int e^x \, dx = e^x + C$$ (exponential rule)

$$\int a^x \, dx = \frac{a^x}{\ln(a)} + C$$ (general exponential rule)

Trigonometric Integrals

$$\int \sin(x) \, dx = -\cos(x) + C$$

$$\int \cos(x) \, dx = \sin(x) + C$$

$$\int \tan(x) \, dx = -\ln|\cos(x)| + C = \ln|\sec(x)| + C$$

$$\int \cot(x) \, dx = \ln|\sin(x)| + C$$

$$\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C$$

$$\int \csc(x) \, dx = \ln|\csc(x) - \cot(x)| + C$$

Integration Rules

$$\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$$ (sum rule)

$$\int cf(x) \, dx = c \int f(x) \, dx$$ (constant multiple rule)

Integration by parts: $$\int u \, dv = uv - \int v \, du$$

Integration by substitution: $$\int f(g(x))g'(x) \, dx = \int f(u) \, du$$ where $$u = g(x)$$

Inverse Trigonometric Integrals

$$\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(x) + C$$

$$\int \frac{-1}{\sqrt{1-x^2}} \, dx = \arccos(x) + C$$

$$\int \frac{1}{1+x^2} \, dx = \arctan(x) + C$$

$$\int \frac{1}{x\sqrt{x^2-1}} \, dx = \text{arcsec}(x) + C$$ for $$|x| > 1$$

Basic Limit Properties

$$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$$

$$\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$$

$$\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$$

$$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$$ if $$\lim_{x \to a} g(x) \neq 0$$

$$\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$$

Special Limits

$$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$$

$$\lim_{x \to 0} \frac{1-\cos(x)}{x} = 0$$

$$\lim_{x \to 0} \frac{e^x-1}{x} = 1$$

$$\lim_{x \to 0} (1+x)^{1/x} = e$$

$$\lim_{x \to \infty} \left(1+\frac{1}{x}\right)^x = e$$

Techniques for Evaluating Limits

Direct Substitution: If $$f$$ is continuous at $$a$$, then $$\lim_{x \to a} f(x) = f(a)$$

Factoring: For algebraic expressions with common factors

Rationalization: For expressions with radicals

L'Hôpital's Rule: For indeterminate forms $$\frac{0}{0}$$ or $$\frac{\infty}{\infty}$$

Squeeze Theorem: If $$g(x) \leq f(x) \leq h(x)$$ and $$\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$$, then $$\lim_{x \to a} f(x) = L$$

Limits at Infinity

For rational functions $$\frac{P(x)}{Q(x)}$$ as $$x \to \infty$$, compare the highest degree terms

$$\lim_{x \to \infty} \frac{a_n x^n + \ldots + a_1 x + a_0}{b_m x^m + \ldots + b_1 x + b_0}$$

If $$n < m$$: Limit is 0

If $$n = m$$: Limit is $$\frac{a_n}{b_m}$$

If $$n > m$$: Limit is $$\infty$$ or $$-\infty$$

Common Sequences

Arithmetic Sequence: $$a_n = a_1 + (n-1)d$$

Geometric Sequence: $$a_n = a_1 \cdot r^{n-1}$$

Fibonacci Sequence: $$F_n = F_{n-1} + F_{n-2}$$ with $$F_1 = F_2 = 1$$

Harmonic Sequence: $$a_n = \frac{1}{n}$$

Square Numbers: $$a_n = n^2$$

Triangular Numbers: $$a_n = \frac{n(n+1)}{2}$$

Sequence Sums

Arithmetic Series: $$\sum_{i=1}^{n} a_i = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$$

Geometric Series: $$\sum_{i=1}^{n} a_i = a_1 \frac{1-r^n}{1-r}$$ for $$r \neq 1$$

Infinite Geometric Series: $$\sum_{i=1}^{\infty} a_i = \frac{a_1}{1-r}$$ for $$|r| < 1$$

Sum of Squares: $$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$$

Sum of Cubes: $$\sum_{i=1}^{n} i^3 = \left[\frac{n(n+1)}{2}\right]^2$$

Convergence Tests

Limit Test: If $$\lim_{n \to \infty} a_n = L$$ exists, the sequence converges to $$L$$

Monotone Convergence Theorem: A bounded monotonic sequence converges

Ratio Test: If $$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = r$$

- If $$r < 1$$, the series converges absolutely

- If $$r > 1$$ or $$r = \infty$$, the series diverges

- If $$r = 1$$, the test is inconclusive

Special Sequences

Catalan Numbers: $$C_n = \frac{1}{n+1}\binom{2n}{n}$$

Bernoulli Numbers: Complex sequence related to sum of powers

Bell Numbers: Number of partitions of a set with n elements

Stirling Numbers: Related to permutations and combinations

Lucas Numbers: Similar to Fibonacci but with $$L_1 = 1, L_2 = 3$$

Common Series

Geometric Series: $$\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}$$ for $$|r| < 1$$

p-Series: $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$ converges if $$p > 1$$, diverges if $$p \leq 1$$

Harmonic Series: $$\sum_{n=1}^{\infty} \frac{1}{n}$$ (diverges)

Alternating Harmonic Series: $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \ln(2)$$

Basel Problem: $$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

Convergence Tests

Comparison Test: Compare with a known convergent/divergent series

Limit Comparison Test: Compare limits of terms

Ratio Test: Examine $$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$$

Root Test: Examine $$\lim_{n \to \infty} \sqrt[n]{|a_n|}$$

Integral Test: Compare series with an integral

Alternating Series Test: For alternating series with decreasing terms

Power Series

Form: $$\sum_{n=0}^{\infty} a_n (x-c)^n$$

Radius of Convergence: Value $$R$$ such that the series converges for $$|x-c| < R$$

Interval of Convergence: Range of $$x$$ values for which the series converges

Term-by-term Differentiation: $$\frac{d}{dx}\sum_{n=0}^{\infty} a_n (x-c)^n = \sum_{n=1}^{\infty} n a_n (x-c)^{n-1}$$

Term-by-term Integration: $$\int \sum_{n=0}^{\infty} a_n (x-c)^n dx = C + \sum_{n=0}^{\infty} \frac{a_n}{n+1} (x-c)^{n+1}$$

Taylor Series

Taylor Series at $$x = c$$: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (x-c)^n$$

Maclaurin Series (Taylor at $$x = 0$$): $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$$

Common Maclaurin Series:

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$$

$$\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots$$

$$\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots$$