if you evaluate the integral of:
t csc^2 t dt, is the answer
-t cot t + ln|sint| + C
Differentiate the answer to see if it is correct.
To evaluate the integral of t*csc^2(t)dt, we can use the technique of integration by parts. Integration by parts is based on the product rule for differentiation.
The formula for integration by parts is:
∫u dv = uv - ∫v du
In this case, we can choose u = t and dv = csc^2(t)dt.
To find du, we differentiate u with respect to t:
du/dt = 1
To find v, we integrate dv with respect to t:
v = ∫csc^2(t)dt
The integral of csc^2(t)dt is a well-known integral. It evaluates to -cot(t) + C1, where C1 is the constant of integration.
Now we have all the components to apply the integration by parts formula:
∫t*csc^2(t)dt = uv - ∫v du
= t*(-cot(t) + C1) - ∫(-cot(t) + C1) dt
= -t*cot(t) + C1*t + ∫cot(t) dt - C1∫dt
= -t*cot(t) + C1*t + ln|sin(t)| - C1t + C2
= -t*cot(t) + ln|sin(t)| + C, where C = C1 + C2
So, the answer to the integral is -t*cot(t) + ln|sin(t)| + C.
To check if this answer is correct, we can differentiate it with respect to t and see if we obtain t*csc^2(t).
Differentiating -t*cot(t) + ln|sin(t)| + C with respect to t, we get:
d/dt (-t*cot(t) + ln|sin(t)| + C) = -cot(t) + (-t*(-csc^2(t))) + 0
= -cot(t) + tcsc^2(t)
Therefore, the differentiated answer matches our original integrand t*csc^2(t). Hence, the answer is correct.