math 2b, section 2

integration by parts formula

∫u dv = uv - ∫v du

exponential(integration by parts)

dv

trigonometric(integration by parts)

dv

algebraic(integration by parts)

u or dv

inverse trig(integration by parts)

u

logarithmic(integration by parts)

u

$$\int\tan(x)\,dx$$

$$-\ln|\cos(x)|+C$$

$$\int\cot(x)\,dx$$

$$\ln|\sin(x)|+C$$

$$\int\sec(x)\,dx$$

$$\ln|\sec(x)+\tan(x)|+C$$

$$\int\csc(x)\,dx$$

$$-\ln|\csc(x)+\cot(x)|+C$$

1. Odd Number of Cosines($\int \sin^n(x)\cos^m(x)\,dx$ )

save one factor of $\cos(x)$ for $du = \cos(x)\,dx$, convert the remaining even powers of cosine using $\cos^2(x) = 1 - \sin^2(x)$, and substitute $u = \sin(x)$.

2. Odd number of sines($\int \sin^n(x)\cos^m(x)\,dx$ )

save one factor of $\sin(x)$ for $du = -\sin(x)\,dx$, convert the remaining even powers of sine using $\sin^2(x) = 1 - \cos^2(x)$, and substitute $u = \cos(x)$.

3. Odd number of tangents/cotangents($\int \tan^n(x)\sec^m(x)\,dx$ )

save a factor of $\sec(x)\tan(x)$ for $du = \sec(x)\tan(x)\,dx$, convert remaining tangents using $\tan^2(x) = \sec^2(x) - 1$, and substitute $u = \sec(x)$.

4. Even Number of Secants/Cosecants($\int \tan^n(x)\sec^m(x)\,dx$ )

save a factor of $\sec^2(x)$ for $du = \sec^2(x)\,dx$, convert remaining secants using $\sec^2(x) = 1 + \tan^2(x)$, and substitute $u = \tan(x)$.

special: Zero Secants and Even Number of Tangents($\int\tan^{n}\left(x\right)\,dx$ )

convert using using $\tan^2(x) = \sec^2(x) - 1$ , if more than 2, substitute $u=\tan(x)$ and du = \sec^2(x)\differentialD x

$$\sin^2(\theta)$$

$$\frac{1 - \cos(2\theta)}{2}$$

$$\cos^2(\theta)$$

$$\frac{1 + \cos(2\theta)}{2}$$

5. Even number of Sines and Cosines

The identities $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$ and $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$ , used to evaluate trigonometric integrals containing an even number of sines and an even number of cosines.

6. Even Tangents and Odd Secants(ex. $\int \sec^3(x)\,dx$ )

Use integration by parts

$$\sqrt{1 - x^2}$$

$$x = \sin(\theta)$$

$$\sqrt{1 + x^2}$$

$$x = \tan(\theta)$$

$$\sqrt{x^2-1}$$

$$x = \sec(\theta)$$

Improper Integral

A definite integral $\int_a^b f(x)\,dx$ in which either the interval of integration is infinite or the integrand $f(x)$ is undefined or discontinuous at some value $x \in [a, b]$.

Type 1 Improper Integral

An improper integral with infinite integration limits, evaluated as a limit of the form $\lim_{T \rightarrow \infty} \int_a^T f(x)\,dx$ or $\lim_{T \rightarrow -\infty} \int_T^b f(x)\,dx$.

Type 2 Improper Integral

An improper integral where the integrand has a vertical asymptote or discontinuity on the interval $[a, b]$, evaluated by taking a one-sided limit toward the point of discontinuity.