Evalute the definite integral (cosx)/(3 sin(x)+15)dx= +C
Correction its a indefinte integral not a definite
To evaluate the definite integral of (cosx)/(3sin(x) + 15)dx, we can use a standard integration technique called substitution.
Step 1: Identify a suitable substitution
Let's substitute u = 3sin(x) + 15. This choice ensures that when we differentiate u with respect to x, we get du/dx = 3cos(x).
Step 2: Calculate du/dx and rearrange to get dx
To find du/dx, differentiate both sides of u = 3sin(x) + 15 with respect to x:
du/dx = 3cos(x)
Now rearrange the equation to solve for dx:
dx = du / 3cos(x)
Step 3: Substitute u and dx in the integral expression
Substituting u = 3sin(x) + 15 and dx = du / 3cos(x) in the given integral, we have:
∫(cosx)/(3sin(x) + 15) dx = ∫(1/u) du
Step 4: Evaluate the new integral
The indefinite integral of 1/u with respect to u is simply ln|u|. Therefore, we have:
∫(1/u) du = ln|u| + C
Step 5: Substitute back for u
Since u = 3sin(x) + 15, we substitute back to get:
ln|3sin(x) + 15| + C
So, the definite integral of (cosx)/(3sin(x) + 15)dx is ln|3sin(x) + 15| + C, where C is the constant of integration.