integrate:cosx+2/cosx+sinx
To integrate the function (cosx + 2)/(cosx + sinx), we can use the technique of u-substitution. Here's how you can do it step by step:
Step 1: Identify the u-substitution
We need to make a choice for u that will simplify the integral. In this case, let's choose u = cosx + sinx. This means that we need to find du/dx in order to express dx in terms of du.
Differentiating u = cosx + sinx with respect to x, we get:
du/dx = -sinx + cosx
Rewriting it in terms of dx, we have:
dx = du / (-sinx + cosx)
Step 2: Rewrite the integral using u
Now, let's substitute the value of dx and u into the integral:
∫ (cosx + 2) / (cosx + sinx) dx = ∫ (cosx + 2) / u (-sinx + cosx) du
Step 3: Simplify the integrand
Using the u-substitution, we can simplify the integrand by replacing cosx + sinx with u:
∫ (cosx + 2) / u (-sinx + cosx) du = ∫ (cosx + 2) / u^2 du
Step 4: Integrate the function
Now, we have a simple rational function that we can integrate. To integrate (cosx + 2) / u^2, we can use the power rule for integration:
∫ (cosx + 2) / u^2 du = ∫ (cosx / u^2) du + ∫ (2 / u^2) du
Using the power rule, the integral of cosx/u^2 is -cosx/u, and the integral of 2/u^2 is -2/u. Therefore, the final result is:
∫ (cosx + 2) / u^2 du = -cosx/u - 2/u + C
Step 5: Substitute u back in terms of x
In the original substitution, we set u = cosx + sinx. Therefore, replacing u back with cosx + sinx gives us the final answer:
∫ (cosx + 2) / (cosx + sinx) dx = -cosx / (cosx + sinx) - 2 / (cosx + sinx) + C
And there you have it! The integral of (cosx + 2)/(cosx + sinx) is -cosx / (cosx + sinx) - 2 / (cosx + sinx) + C, where C is the constant of integration.