What is f(x) = int -cos^3x +tanx dx if f(pi)=-2 ?
1 Answer
Mar 20, 2016
Explanation:
This can be split into two integrals:
f(x)=-intcos^3xdx+inttanxdx
The second integral
To find the first, do what follows:
-intcos^3xdx=-intcos^2xcosxdx
=-int(1-sin^2x)cosxdx
Here, use substitution: let
=-int1-u^2du
Which we can integrate using
=-u+u^3/3+C=-sinx+sin^3x/3+C
Thus, the whole expression equals
f(x)=-sinx+sin^3x/3-lnabscosx+C
Since
-2=-sinpi+sin^3pi/3-lnabscospi+C
-2=-0+0^3/3-lnabs(-1)+C
-2=0+0+ln1+C
-2=C
Thus, we obtain
f(x)=-sinx+sin^3x/3-lnabscosx-2