Can someone help me to integrate by substitution the indefinite integral:
(sinx)/ (1+cos(^2)x)?
To integrate the given expression, we can use the technique of substitution. Here's how to do it step by step:
1. Let's start by identifying the part of the integral that we will substitute. In this case, it's the expression (1 + cos^2(x)).
2. Let's substitute u for this expression:
u = 1 + cos^2(x)
3. Now, we need to find the derivative of u with respect to x.
du/dx = d/dx(1 + cos^2(x))
4. To find the derivative, we apply the chain rule:
du/dx = 0 + 2cos(x)(-sin(x))
= -2cos(x)sin(x)
5. Next, we solve the equation from step 2 for cos^2(x):
u = 1 + cos^2(x)
cos^2(x) = u - 1
6. Now, we substitute back into the integral:
∫ (sin(x)/(1 + cos^2(x))) dx
= ∫ (sin(x)/(u - 1)) dx
7. To complete the substitution, we need to replace dx with an expression involving du.
Recall from step 4 that du/dx = -2cos(x)sin(x).
Rearranging, we have dx = -du/(2cos(x)sin(x)).
Substituting this into the integral gives:
= ∫ (sin(x)/(u - 1)) (-du/(2cos(x)sin(x)))
8. Now, we can simplify the expression:
= -1/2 ∫ du/(u - 1)
9. The remaining integral is a straightforward one:
= -1/2 ∫ du/u (integrating with respect to u)
10. The integral of du/u is the natural logarithm of the absolute value of u:
= -1/2 ln|u| + C
11. Finally, substitute back the expression for u:
= -1/2 ln|1 + cos^2(x)| + C
So, the indefinite integral of (sin(x)/(1 + cos^2(x))) is equal to -1/2 ln|1 + cos^2(x)| + C, where C is the constant of integration.