Integration Formulas

Vocabulary flashcards covering standard integration formulas for exponential, logarithmic, rational, and trigonometric functions.

$$\int e^x\,dx$$

$$e^x + C$$

$$\int x^{-1}\,dx$$

$$\ln\|x\| + C$$

$$\int e^{ax}\,dx$$

$$\frac{1}{a}e^{ax} + C$$

$$\int a^x\,dx$$

$$\frac{a^x}{\ln(a)} + C$$

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

$$\sin(x) + C$$

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

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

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

$$\sec(x) + C$$

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

$$-\csc(x) + C$$

$$\int \sec^2(x)\,dx$$

$$\tan(x) + C$$

$$\int \csc^2(x)\,dx$$

$$-\cot(x) + C$$

$$\int \frac{dx}{x^2 + a^2}$$

$$\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$$

$$\int \frac{x\,dx}{x^2 + a^2}$$

$$\frac{1}{2}\ln\|x^2 + a^2\| + C$$

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

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

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

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

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

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

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

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

Pythagorean Identity for sine and cosine

$$\sin^2(x) + \cos^2(x) = 1$$

Double Angle Identity for $\sin(2x)$

$$2\sin(x)\cos(x)$$

Double Angle Identity for $\cos(2x)$

$$\cos^2(x) - \sin^2(x)$$

Power Reduction Identity for $\cos^2(x)$

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

Power Reduction Identity for $\sin^2(x)$

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

Product to Sum Identity for $\sin(A)\cos(B)$

$$\frac{1}{2}[\sin(A-B) + \sin(A+B)]$$

Product to Sum Identity for $\sin(A)\sin(B)$

$$\frac{1}{2}[\cos(A-B) - \cos(A+B)]$$

Product to Sum Identity for $\cos(A)\cos(B)$

$$\frac{1}{2}[\cos(A-B) + \cos(A+B)]$$

Pythagorean Identity for $\tan(x)$ and $\sec(x)$

$$1 + \tan^2(x) = \sec^2(x)$$

Pythagorean Identity for $\cot(x)$ and $\csc(x)$

$$1 + \cot^2(x) = \csc^2(x)$$

Integration strategy when both powers on sine and cosine are even

Use Power Reduction Identities.

Integration strategy when the power on sine is odd

Use $u$-substitution with $u = \cos(x)$ and the Pythagorean Identity.

Integration strategy when the power on cosine is odd

Use $u$-substitution with $u = \sin(x)$ and the Pythagorean Identity.

Integration strategy when the power on secant is even

Use $u$-substitution with $u = \tan(x)$ and the Pythagorean Identity.

Integration strategy when the power on cosecant is even

Use $u$-substitution with $u = \cot(x)$ and the Pythagorean Identity.

Integration strategy when the power on tangent is odd

Use $u$-substitution with $u = \sec(x)$ and the Pythagorean Identity.

Integration strategy when the power on cotangent is odd

Use $u$-substitution with $u = \csc(x)$ and the Pythagorean Identity.

Integration strategy when the power on tangent is even

Factor out $\tan^2(x)$, use $\tan^2(x) = \sec^2(x) - 1$, and integrate in terms of powers of secant.