The integral of every common function

Integral of a constant \int k\,dx = kx + C The integral of a constant k is kx + C, and the integral of 1 is x + C. See why, the area picture, definite integrals and the mistakes to avoid. Integral of x \int x\,dx = \frac{x^2}{2} + C The integral of x is x²/2 + C. See why with the reverse power rule and a triangle, plus definite integrals, the linear form ax + b and common mistakes. Integral of x² \int x^2\,dx = \frac{x^3}{3} + C The integral of x² is x³/3 + C. See the reverse power rule, a Riemann sum proof, definite integrals, the (ax + b)² form and the mistakes students make. Integral of √x \int \sqrt{x}\,dx = \frac{2}{3}x^{3/2} + C The integral of √x is (2/3)x^(3/2) + C. Rewrite the root as a power, check by differentiating, and see √(ax + b), 1/√x and definite integral examples. Integral of 1/x \int \frac{1}{x}\,dx = \ln\lvert x\rvert + C The integral of 1/x is ln|x| + C. See why the power rule fails here, where the absolute value comes from, and how to handle 1/(ax + b) and f'/f. Integral of 1/x² \int \frac{1}{x^2}\,dx = -\frac{1}{x} + C The integral of 1/x² is -1/x + C. Rewrite it as x^(-2), check by differentiating, and see improper integrals like 1 to infinity and the classic -1 to 1 trap. Integral of |x| \int \lvert x\rvert\,dx = \frac{x\lvert x\rvert}{2} + C The integral of |x| is x|x|/2 + C. See how to split at zero, why the answer is not |x²/2|, and how to evaluate definite integrals like |x - 1| from 0 to 4. Integral of x/(x² + 1) \int \frac{x}{x^2 + 1}\,dx = \frac{1}{2}\ln\left(x^2 + 1\right) + C The integral of x/(x² + 1) is ½ ln(x² + 1) + C. See the u-substitution step by step, why no absolute value is needed, and related integrals like 1/(x² + 1). Integral of 1/(x² - 1) \int \frac{1}{x^2 - 1}\,dx = \frac{1}{2}\ln\left\lvert\frac{x - 1}{x + 1}\right\rvert + C The integral of 1/(x² - 1) is ½ ln|(x - 1)/(x + 1)| + C. See the partial fraction steps, the general 1/(x² - a²) formula and worked definite integrals.

Integral of sin x \int \sin x\,dx = -\cos x + C The integral of sin x is -cos x + C. See where the minus sign comes from, the area of one arch, sin(ax + b) by substitution and worked definite integrals. Integral of cos x \int \cos x\,dx = \sin x + C The integral of cos x is sin x + C. See why there is no minus sign, the area of one hump, cos(ax + b) by substitution and worked definite integral examples. Integral of tan x \int \tan x\,dx = \ln\lvert\sec x\rvert + C The integral of tan x is ln|sec x| + C, or -ln|cos x| + C. See the u-substitution with u = cos x, why both forms agree, and worked definite integrals. Integral of sec x \int \sec x\,dx = \ln\lvert\sec x + \tan x\rvert + C The integral of sec x is ln|sec x + tan x| + C. See the multiply-by-one trick step by step, the equivalent forms, a check by differentiating and examples. Integral of csc x \int \csc x\,dx = -\ln\lvert\csc x + \cot x\rvert + C The integral of csc x is -ln|csc x + cot x| + C, also written ln|csc x - cot x| or ln|tan(x/2)|. See the derivation, why the forms agree and examples. Integral of cot x \int \cot x\,dx = \ln\lvert\sin x\rvert + C The integral of cot x is ln|sin x| + C. See the u-substitution with u = sin x, how it pairs with tan x, plus cot(ax), cot² x and definite integral examples. Integral of sec² x \int \sec^2 x\,dx = \tan x + C The integral of sec² x is tan x + C. See why it follows from the derivative of tan x, the sec²(ax) form, related integrals like tan² x and csc² x, and examples. Integral of sin² x \int \sin^2 x\,dx = \frac{x}{2} - \frac{\sin 2x}{4} + C The integral of sin² x is x/2 - sin(2x)/4 + C. See the power-reducing identity step by step, a check by differentiating, sin²(ax) and definite integrals. Integral of cos² x \int \cos^2 x\,dx = \frac{x}{2} + \frac{\sin 2x}{4} + C The integral of cos² x is x/2 + sin(2x)/4 + C. See the power-reducing identity, a check, cos²(ax) and the area of a circle. Integral of tan² x \int \tan^2 x\,dx = \tan x - x + C The integral of tan² x is tan x - x + C. See how the identity tan² x = sec² x - 1 does all the work, plus tan²(ax), tan³ x and definite integral examples. Integral of sec³ x \int \sec^3 x\,dx = \frac{1}{2}\left(\sec x\tan x + \ln\lvert\sec x + \tan x\rvert\right) + C The integral of sec³ x is ½(sec x tan x + ln|sec x + tan x|) + C. See the integration by parts loop step by step, a full check and the arc length application. Integral of sec x tan x \int \sec x\tan x\,dx = \sec x + C The integral of sec x tan x is sec x + C. See why it reverses the derivative of sec x, how to spot it in disguise as sin x/cos² x, and substitution examples. Integral of sin 2x \int \sin 2x\,dx = -\frac{1}{2}\cos 2x + C The integral of sin 2x is -cos(2x)/2 + C. See the u-substitution, why sin² x + C and -cos² x + C are also correct, and worked definite integral examples.

Integral of 1/(1+x²) \int \frac{1}{1+x^2}\,dx = \arctan x + C The integral of 1/(1+x²) is arctan x + C. See why with a tan substitution, the 1/(a²+x²) version, a graph and worked examples. Integral of 1/√(1−x²) \int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C The integral of 1/√(1−x²) is arcsin x + C. See the sine substitution, the 1/√(a²−x²) version, a graph, definite integrals and examples. Integral of √(1−x²) \int \sqrt{1-x^2}\,dx = \frac{1}{2}\left(x\sqrt{1-x^2} + \arcsin x\right) + C The integral of √(1−x²) is ½(x√(1−x²) + arcsin x) + C. Learn the trig substitution, the link to the area of a circle and worked examples. Integral of arctan x \int \arctan x\,dx = x\arctan x - \frac{1}{2}\ln(1+x^2) + C The integral of arctan x is x arctan x − ½ ln(1+x²) + C. See the integration by parts steps, a check, definite integrals and examples. Integral of arcsin x \int \arcsin x\,dx = x\arcsin x + \sqrt{1-x^2} + C The integral of arcsin x is x arcsin x + √(1−x²) + C. Learn the integration by parts steps, the arccos version, a graph and worked examples.

Integral of eˣ \int e^x\,dx = e^x + C The integral of eˣ is eˣ + C. See why eˣ is its own antiderivative, a graph of the area, definite and improper integrals and worked examples. Integral of e²ˣ \int e^{2x}\,dx = \frac{1}{2}e^{2x} + C The integral of e^(2x) is ½e^(2x) + C, and in general ∫e^(kx) dx = (1/k)e^(kx) + C. See the substitution, a graph and worked examples. Integral of e⁻ˣ \int e^{-x}\,dx = -e^{-x} + C The integral of e^(−x) is −e^(−x) + C. See the substitution, why the area from 0 to infinity is 1, a graph and examples like x e^(−x). Integral of e^(x²) \int e^{-x^2}\,dx = \frac{\sqrt{\pi}}{2}\,\operatorname{erf}(x) + C e^(x²) and e^(−x²) have no elementary antiderivative. Learn the erf and erfi functions, the power series answer and why ∫e^(−x²) over ℝ is √π. Integral of 2ˣ \int 2^x\,dx = \frac{2^x}{\ln 2} + C The integral of 2ˣ is 2ˣ/ln 2 + C, and ∫aˣ dx = aˣ/ln a + C. See the derivation from e^(x ln a), a graph, definite integrals and examples. Integral of ln x \int \ln x\,dx = x\ln x - x + C The integral of ln x is x ln x − x + C. See the integration by parts steps, a check by differentiating, a graph and examples like ln(2x) and (ln x)². Integral of 1/(x ln x) \int \frac{1}{x\ln x}\,dx = \ln\lvert\ln x\rvert + C The integral of 1/(x ln x) is ln|ln x| + C. See the u = ln x substitution, why its improper integral diverges, a graph and worked examples.

Integral of x eˣ \int x e^x\,dx = (x - 1)e^x + C The integral of x eˣ is (x − 1)eˣ + C. See the integration by parts steps, why u = x is the right choice, a graph and examples like x e^(2x). Integral of x ln x \int x\ln x\,dx = \frac{x^2}{2}\ln x - \frac{x^2}{4} + C The integral of x ln x is (x²/2) ln x − x²/4 + C. See the integration by parts steps with u = ln x, a check, a graph and the xⁿ ln x formula. Integral of x sin x \int x\sin x\,dx = \sin x - x\cos x + C The integral of x sin x is sin x − x cos x + C. See the integration by parts steps, a check by differentiating, a graph and examples like x² sin x. Integral of x cos x \int x\cos x\,dx = x\sin x + \cos x + C The integral of x cos x is x sin x + cos x + C. See the integration by parts steps, a check, a graph, definite integrals and examples like x² cos x. Integral of eˣ sin x \int e^x\sin x\,dx = \frac{e^x}{2}\left(\sin x - \cos x\right) + C The integral of eˣ sin x is ½eˣ(sin x − cos x) + C. Learn the integrate-twice-and-solve trick, a check, a graph and the e^(ax) sin(bx) formula. Integral of x² eˣ \int x^2 e^x\,dx = (x^2 - 2x + 2)e^x + C The integral of x² eˣ is (x² − 2x + 2)eˣ + C. See integration by parts twice, the tabular shortcut, a check, a graph and the xⁿ eˣ pattern.