6) Evaluate the indefinite integral.
∫cosx/(7sinx+35)dx
let u = 7sinx+35
du = 7cosx
and the integrand now becomes
1/7 du/u
To evaluate the indefinite integral ∫cos(x)/(7sin(x) + 35) dx, we can start by making a substitution. Let u = 7sin(x) + 35. Then, du = 7cos(x) dx.
Next, we can rewrite the integral in terms of u:
∫(cos(x))/(7sin(x) + 35) dx = ∫(1/7)(1/u) du
Now, let's simplify the expression:
∫(1/7)(1/u) du = (1/7)∫du/u
Using the power rule for integration, the integral becomes:
(1/7) ln|u| + C
Finally, substituting back u = 7sin(x) + 35, we have:
(1/7) ln|7sin(x) + 35| + C
Therefore, the indefinite integral of cos(x)/(7sin(x) + 35) dx is (1/7) ln|7sin(x) + 35| + C, where C is the constant of integration.
To evaluate the indefinite integral ∫cosx/(7sinx+35)dx, we can use a technique called substitution.
Step 1: Simplify the integral
To simplify the integral, we can divide both the numerator and denominator of the integrand by 7. This gives us:
∫cosx/(7sinx+35)dx = ∫(cosx/7)/(sinx+5)dx
Step 2: Choose a substitution
Now, let's choose a substitution to make the integral easier to solve. We can let u = sinx+5. This choice simplifies the integral even further.
Step 3: Find the derivative du/dx
To find du/dx, the derivative of u with respect to x, we differentiate both sides of the equation u = sinx+5 with respect to x.
The derivative of sinx is cosx, and the derivative of a constant (5 in this case) is 0. So, du/dx = cosx.
Step 4: Rewrite the integral in terms of u and du
Now that we have our substitution and the derivative of u with respect to x, we can rewrite the integral in terms of u and du.
∫(cosx/7)/(sinx+5)dx = ∫(1/7)/(sinx+5) * cosx dx
= ∫(1/7)/(u) du
Step 5: Solve the new integral
Now we can solve the integral with respect to u:
∫(1/7)/(u) du = (1/7) ∫(1/u) du
= (1/7) ln|u| + C
Step 6: Substitute back for u
Finally, substitute back for u using the substitution we made earlier:
(1/7) ln|u| + C = (1/7) ln|sinx+5| + C
So, the indefinite integral of ∫cosx/(7sinx+35)dx is (1/7) ln|sinx+5| + C, where C is the constant of integration.