What is f(x) = int 3x-2cscx dx if f((5pi)/4) = 0 ?

1 Answer
Jan 11, 2018

f(x) = 3/2x^2+2ln|csc(x)+cot(x)|-21.3691

Explanation:

int3x-2cscxdx

=int3xdx-2intcscxdx

To integrate the second integral re write and use the following substitution:

intcscxdx=intcscx(cscx+cotx)/(cscx+cotx)dx

u=cscx+cotx
-> du=-cotxcscx-csc^2xdx=-cscx(cscx+cotx)dx

Now substitute into the integral:

intcscx(cscx+cotx)/(cscx+cotx)dx=-int(du)/u=-lnu+C_0

Reverse the substitution:

-ln|u| +C_0 = -ln|csc(x)+cot(x)|+C_0

So, returning to the original integral:

int3xdx-2intcscxdx=3/2x^2+2ln|csc(x)+cot(x)|+C

Now solve for C.

f((5pi)/4)
=3/2((5pi)/4)^2+2ln|csc((5pi)/4)+cot((5pi)/4)|+C=0

75/32pi^2+2ln(-sqrt(2)+1)+C

->21.3691+C=0-> C=-21.3691

f(x) = 3/2x^2+2ln|csc(x)+cot(x)|-21.3691