1 Integration Techniques Free Online FlashCards

What is an antiderivative?

An antiderivative of $f$ is a function $F$ satisfying $F'(x)=f(x)$ on the interval.

Why does an indefinite integral include $+C$?

The indefinite integral represents all antiderivatives: $\int f(x)\,dx=F(x)+C$, where $C$ is an arbitrary constant.

State the power rule for integration.

For $n\ne -1$, $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$. The case $n=-1$ instead gives $\int \frac{1}{x}\,dx=\ln|x|+C$.

What does a definite integral represent?

A definite integral is signed accumulation: regions below the $x$-axis contribute negatively. If $f(x)\ge 0$ on $[a,b]$, it equals the geometric area under the graph.

State Part 1 of the Fundamental Theorem of Calculus.

If $F(x)=\int_a^x f(t)\,dt$ and $f$ is continuous, then $F'(x)=f(x)$.

State Part 2 of the Fundamental Theorem of Calculus.

If $F' = f$ on $[a,b]$, then $\int_a^b f(x)\,dx=F(b)-F(a)$. This evaluates a definite integral from any antiderivative.

What is the purpose of $u$-substitution?

Choose $u=g(x)$, compute $du=g'(x)\,dx$, rewrite the integral in $u$, integrate, and substitute back. This reverses the chain rule.

State the integration-by-parts formula.

Integration by parts uses $\int u\,dv=uv-\int v\,du$. It is based on reversing the product rule and is useful for suitable products.

How are odd powers handled in sine–cosine integrals?

For $\int \sin^m x\cos^n x\,dx$, save one sine factor when $m$ is odd or one cosine factor when $n$ is odd, then use a Pythagorean identity.

Match each radical form to its trigonometric substitution.

Use $x=a\sin\theta$ for $\sqrt{a^2-x^2}$, $x=a\tan\theta$ for $\sqrt{a^2+x^2}$, and $x=a\sec\theta$ for $\sqrt{x^2-a^2}$.

When is partial-fraction decomposition applicable?

Partial fractions applies to rational functions after polynomial division if needed. Distinct linear factors receive constant numerators; an irreducible quadratic receives a linear numerator.

What is the midpoint rule formula?

With $\Delta x=\frac{b-a}{n}$, the midpoint rule is $M_n=\Delta x\sum_{i=1}^n f\left(\frac{x_{i-1}+x_i}{2}\right)$.