To calculate an integral, you can follow these general steps:
Identify the function you want to integrate. Let's use an example: f(x) = 2x^2 + 3x + 1.
Determine the limits of integration. This specifies the range over which you want to find the integral. For example, let's integrate from x = 0 to x = 5.
Apply the power rule of integration. This rule states that the integral of x^n with respect to x is (1/(n+1)) * x^(n+1). For each term in the function, increase the exponent by 1 and divide the coefficient by the new exponent.
In our example, we have: ∫(2x^2 + 3x + 1) dx = (2/3) * x^3 + (3/2) * x^2 + x + C
Note: C represents the constant of integration, which is added to account for any possible constant terms that may have been present in the original function.
Evaluate the integral between the limits of integration. Substitute the upper limit of integration (in this case, x = 5) into the integrated function, and subtract the result of substituting the lower limit of integration (x = 0). This will give you the value of the definite integral.
In our example, the definite integral would be: ∫[0,5] (2x^2 + 3x + 1) dx = [(2/3) * 5^3 + (3/2) * 5^2 + 5] - [(2/3) * 0^3 + (3/2) * 0^2 + 0]
Simplifying the expression will give you the numerical value of the definite integral.
Note that this is a basic example, and integrals can become more complex depending on the function being integrated.
The (unbounded) cosine primitive is equal to a positive sine, and the sine primitive is equal to a negative cosine. ∫ (cosine) = sine which gives: ∫ (cos (x)) dx = sin (x) ∫ (sine) = - cosine which gives: ∫ (sin (x)) dx = - cos (x)
You have to imagine being in the same story, but this time the scene takes place when SINUS arrived on dry land (he is positive and happy to have left the water) ! Now that he's safe, we put his head back (we put him in)! When SINUS is integrated, it finds its head (its "CO") and (re) transforms into negative COSINUS! (Negative because eventually he got used to his SINUS, and is not happy with this transformation)!
Use Pythagorean Identities: sin²x=1/2*(1 - cos(2x))/2
∫1/2 -((cos(2x))/2)dx
Use Sum Rule: ∫f(x)+g(x) dx=∫f(x) dx+ ∫g(x) dx.
∫1/2 dx - ∫((cos(2x))/2)dx
Use this rule:∫a dx = ax + C.
(x/2)-∫((cos2x/2))/2 dx
Use Constant Factor Rule ∫c f(x) dx = c ∫f(x) dx.
(x/2)-(1/2) ∫ cos2x dx
Use Integration by Substitution on ∫cos2x dx.
Let u=2x, du=2 dx, then dx=1/2 du
Using u and du above, rewrite ∫cos2x dx.
∫ (cos u)/2 du
Use Constant Factor Rule: ∫cf(x) dx = c∫f(x) dx.
1/2∫cos u du
Use Trigonometric Integration: the integral of cos u is sin u.
(sin u)/2
Substitute u=2x back into the original integral.
(sin2x)/2
Rewrite the integral with the completed substitution.
(x/2)-(sin2x)/4
Add constant.
(x/2)-(sin2x)/4 + C