Question

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

Answer 1

`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`