Solve trigonometric integrals symbolically. This calculator supports sine, cosine, tangent, cotangent, secant and cosecant, common trigonometric identities, powers, linear arguments, substitution patterns, Fresnel integrals, derivative verification and numerical graphing.
Examples can be entered with or without explicit multiplication:
3x, 3*x, sin(3x),
2sin(x).
| Integral | Antiderivative |
|---|---|
| \(\int\sin x\,dx\) | \(-\cos x+C\) |
| \(\int\cos x\,dx\) | \(\sin x+C\) |
| \(\int\tan x\,dx\) | \(-\ln|\cos x|+C\) |
| \(\int\cot x\,dx\) | \(\ln|\sin x|+C\) |
| \(\int\sec x\,dx\) | \(\ln|\sec x+\tan x|+C\) |
| \(\int\csc x\,dx\) | \(-\ln|\csc x+\cot x|+C\) |
| \(\int\sec^2x\,dx\) | \(\tan x+C\) |
| \(\int\csc^2x\,dx\) | \(-\cot x+C\) |
| \(\int\sec x\tan x\,dx\) | \(\sec x+C\) |
| \(\int\csc x\cot x\,dx\) | \(-\csc x+C\) |
Several difficult-looking trigonometric integrals become much simpler after applying an identity.
These identities are especially useful when integrating powers of sine, cosine, tangent and cotangent.
Start by checking whether the integrand is directly related to a standard derivative. For example:
\[ \frac{d}{dx}\sin x=\cos x \] and \[ \frac{d}{dx}\tan x=\sec^2x. \]For an expression such as \[ \cos(5x), \] the inner derivative is \(5\), so \[ \int\cos(5x)\,dx = \frac{1}{5}\sin(5x)+C. \]
For example, \[ \int x\cos(x^2)\,dx \] suggests \[ u=x^2,\qquad du=2x\,dx. \] Therefore, \[ \int x\cos(x^2)\,dx = \frac12\sin(x^2)+C. \]
For \[ \int\sin^2x\,dx, \] the power-reduction identity gives \[ \sin^2x=\frac{1-\cos(2x)}2. \]
A symbolic antiderivative is an exact mathematical expression such as \[ -\cos x+C. \]
Numerical integration instead approximates an integral using numerical values. The graph on this page uses numerical cumulative integration only for visualization.
For definite numerical calculations, see the Definite Integral Calculator and Numerical Integration Calculator.
Trigonometric functions can have discontinuities. For example, \(\tan x\), \(\sec x\), \(\cot x\) and \(\csc x\) are undefined at specific values of \(x\).
For example:
\[ \tan x=\frac{\sin x}{\cos x} \] so tangent is undefined whenever \[ \cos x=0. \]The graph therefore breaks the numerical curve when it encounters undefined or extremely large values instead of drawing a misleading line through an asymptote.
A trigonometric integral is an integral containing one or more trigonometric functions, including sine, cosine, tangent, cotangent, secant and cosecant.
\[ \int\sin(x)\,dx=-\cos(x)+C. \]
\[ \int\cos(x)\,dx=\sin(x)+C. \]
\[ \int\sec^2(x)\,dx=\tan(x)+C. \]
Use the power-reduction identity \[ \sin^2x=\frac{1-\cos(2x)}2. \] This converts the squared sine into a constant and a cosine term that can be integrated directly.
Use \[ \tan^2x=\sec^2x-1. \] Therefore, \[ \int\tan^2x\,dx=\tan x-x+C. \]
Integrals such as \[ \int\sin(x^2)\,dx \] and \[ \int\cos(x^2)\,dx \] do not have elementary antiderivatives. Fresnel functions provide standard exact special-function representations.
The constant \(C\) represents an arbitrary constant. All functions that differ only by a constant have the same derivative.
Summe.org provides free mathematical calculators for calculus, algebra, number theory, summation, derivatives, integrals and related subjects.
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