CalculusCollege2 topic explainers

Calculus 2 course

Integration techniques, applications of integrals and infinite series.

Topics in this course

1.Integrals
Integration runs differentiation in reverse and also measures accumulation. An antiderivative $F$ of $f$ satisfies $F' = f$ , and the indefinite integral $\int f(x)\,dx = F(x) + C$ records the whole family, since constants vanish when differentiated. The definite integral $\int_a^b f(x)\,dx$ is the signed area between the graph and the $x$ -axis from $a$ to $b$ .
- Power rule: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \ne -1$ , and $\int \frac{1}{x}\,dx = \ln|x| + C$ .
- Substitution reverses the chain rule; integration by parts reverses the product rule.
- Area below the axis counts as negative in a definite integral.
Quick check: Evaluate $\int_0^2 4x\,dx$ .
$8$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study integrals
2.Series and Convergence
An infinite series adds infinitely many terms, $\sum_{n=1}^{\infty} a_n$ . It converges if its partial sums approach a finite limit. The geometric series $\sum ar^{n}$ is the model case: it converges to $\frac{a}{1 - r}$ exactly when $|r| < 1$ , which is why $1 + \frac{1}{2} + \frac{1}{4} + \cdots = 2$ .
- $n$ th-term test: if $a_n \not\to 0$ , the series diverges.
- $p$ -series $\sum \frac{1}{n^p}$ converges exactly when $p > 1$ .
- Ratio test: $\lim \left|\frac{a_{n+1}}{a_n}\right| < 1$ means absolute convergence.
Quick check: Sum $\sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n$ .
$2$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study series and convergence

Topics in this course

1.Integrals

2.Series and Convergence

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