Trigonometric 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.