If a function f(x) is continuous on a given interval, then the method of integration gives the area under graph of the particular function f(x). Integral of a function is in fact the anti-derivative of the function and Integral calculator is a tool used to find the integral of the function with respect to a variable.
Example 1:Find the integration of the function, f(x) = 7x3 + 4x2 – 5x + 2.
In order to find the integral, the Power Rule says:
∫xn dx= x(n+1)/ (n+1) + c where ‘c’ is a constant!
Following this rule, we can apply the above formula for every exponent in the function.
∫(fx) dx = 7* x3+1/(3+1) + 4* x2+1/(2+1) –5* x1+1/(1+1) + 2*x + c
∫ f(x) dx = 7x4/ 4 + 4x3/ 3 – 5x2/ 2 + 2x + c
Example 2: Find the anti- derivative of the function, f(x) = sin (2x).
Anti-derivative is the same meaning as finding integral of a function.
Here we have a trigonometric function and we know the fact that derivative of the trigonometric function cos(x) is –sin(x), so we have the opposite interpretation for the integral of it.
So the anti-derivative of sin(x) function is – cos(x).
∫ sin(x) dx = - cos(x) + c
Now ∫ (sin2x) dx = - cos( 2x) / 2 + c