Online Integral Calculator

What are integrals?

Integration is an important tool in calculus that can give an antiderivative or represent area under a curve.

The indefinite integral of f (x)f x, denoted Start Integral, Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Integral Endf xdx, is defined to be the antiderivative of f (x)f x. In other words, the derivative of Start Integral, Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Integral Endf xdx is f (x)f x. Since the derivative of a constant is 0, indefinite integrals are defined only up to an arbitrary constant. For example, Start Integral, Start integrand, Start Sine, Start angle, x , angle End , Sine End , integrand End,Start first variable, x , first variable End , Integral End = - Start Cosine, Start angle, x , angle End , Cosine End +constantsinxdx = -cosx+constant, since the derivative of − Start Cosine, Start angle, x , angle End , Cosine End + constant−cosx + constant is Start Sine, Start angle, x , angle End , Sine Endsinx. The definite integral of f (x)f x from x = ax = a to x = bx = b, denoted Start Definite Integral, Start first lower limit, a , first lower limit End,Start first upper limit, b , first upper limit End,Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Definite Integral Endbaf xdx, is defined to be the signed area between f (x)f x and the xx axis, from x = ax = a to x = bx = b.

Both types of integrals are tied together by the fundamental theorem of calculus. This states that if f (x)f x is continuous on [a,b]a,b and F(x)Fx is its continuous indefinite integral, then Start Definite Integral, Start first lower limit, a , first lower limit End,Start first upper limit, b , first upper limit End,Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Definite Integral End = F(b) - F(a)baf xdx = Fb - Fa. This means Start Definite Integral, Start first lower limit, 0 , first lower limit End,Start first upper limit, π , first upper limit End,Start integrand, Start Sine, Start angle, x , angle End , Sine End , integrand End,Start first variable, x , first variable End , Definite Integral End = (- Start Cosine, Start angle, π , angle End , Cosine End ) - (- Start Cosine, Start angle, 0 , angle End , Cosine End ) = 2π0sinxdx = -cosπ - -cos0 = 2. Sometimes an approximation to a definite integral is desired. A common way to do so is to place thin rectangles under the curve and add the signed areas together. Wolfram|Alpha can solve a broad range of integrals ..