What is f(x) = int x^3- csc4x dx if f(pi/12)=-1 ?

1 Answer
Nov 24, 2017

f(x) ~~ 1/4 x^4 + 1/4(ln(cot(4x) - csc(4x))) -1.1362

Explanation:

int x^3 dx - int csc(4x) dx

1/4 x^4 - int 1/sin(4x) dx

The table of integration tells us that int 1/sin(x) = -ln(cot(x) + csc(x))

Using a u substitution, let u=4x. Then du = 4 dx and 1/4 du = dx

1/4 int 1/sin(u) du = -1/4(ln(cot(u) - csc(u))) + C

Plugging back in gives

f(x) = 1/4 x^4 + 1/4(ln(cot(4x) - csc(4x))) + C

Given the condition that f(pi/12)=-1, we can determine C.

1/4 (pi/12)^4 + 1/4(ln(cot(4 pi/12) - csc(4 pi/12))) + C=-1

pi^4/82944 + 1/4(ln(1/sqrt(3) - 2/sqrt(3))) + C=-1

Given how ugly this is, it makes sense to plug it into a calculator.

-0.1362 + C ~~ -1

C ~~ -1.1362

Thus,

f(x) ~~ 1/4 x^4 + 1/4(ln(cot(4x) - csc(4x))) -1.1362