the question asks to integrate cos(t)/(5sin(t)+8)^2 the answer i got was
xcos(t)/(5sin(t)+8)^2 i think the x is wrong but i am not sure what i am doing wrong...help please.
To integrate the given expression, cos(t)/(5sin(t)+8)^2, you need to use a substitution method. Let's break down the steps to solve the integral.
1. Start by letting u = 5sin(t) + 8. This substitution is chosen because it differentiates to du = 5cos(t)dt.
2. Now, find the derivative of u with respect to t. Since u = 5sin(t) + 8, differentiate both sides of this equation: du/dt = 5cos(t).
3. Rearrange the derivative equation to solve for dt: dt = du / (5cos(t)).
4. Substitute the values of u and dt into the original integral. The integral becomes:
∫ cos(t)/(5sin(t)+8)^2 dt = ∫ (1/u^2) du.
5. Simplify the integral as ∫ du / u^2.
6. Integrate the simplified expression. The integral of du / u^2 is -1/u, which gives us -1/(5sin(t) + 8).
Therefore, the final answer to the integral is -1/(5sin(t) + 8) + C, where C is the constant of integration.
The x in your answer may be incorrect because there is no x term present in the integral expression given. The correct answer is -1/(5sin(t) + 8) + C.