What is int cos^2xsinx-tan^2xcotx dxcos2xsinxtan2xcotxdx?

1 Answer
May 22, 2018

int cos^2(x)sin(x)-tan^2(x)cot(x) dx = lnabs(cos(x))-cos^3(x)/x+Ccos2(x)sin(x)tan2(x)cot(x)dx=ln|cos(x)|cos3(x)x+C

Explanation:

Split the integral up into two

int cos^2(x)sin(x)-tan^2(x)cot(x) dx = int cos^2(x)sin(x)dx - int tan^2(x)cot(x) dxcos2(x)sin(x)tan2(x)cot(x)dx=cos2(x)sin(x)dxtan2(x)cot(x)dx

First Step

For int cos^2(x)sin(x)dxcos2(x)sin(x)dx,

Let u=cos(x)u=cos(x) and thus du=-sin(x)dxdu=sin(x)dx

Substituting, you get

int cos^2(x)sin(x)dx=-int cos^2(x)(-sin(x)dx)=-int u^2du=-u^3/3+C =-cos^3(x)/3+Ccos2(x)sin(x)dx=cos2(x)(sin(x)dx)=u2du=u33+C=cos3(x)3+C

Second Step

For int tan^2(x)cot(x) dxtan2(x)cot(x)dx,

Since tan^2(x)cot(x)=tan^2(x)/tan(x)=tan(x)tan2(x)cot(x)=tan2(x)tan(x)=tan(x)

int tan^2(x)cot(x)=int tan(x)dxtan2(x)cot(x)=tan(x)dx

At this point, you can use a formula sheet to get the answer directly, but if you are interested, you can follow along the next few steps.

int tan(x)dx = int sin(x)/cos(x) dxtan(x)dx=sin(x)cos(x)dx

Now if you let w=cos(x)w=cos(x), then dw=-sin(x)dxdw=sin(x)dx

int sin(x)/cos(x) dx = -int 1/cos(x) * -sin(x)dx=-int 1/w dw = -lnabs(w)+C=lnabs(cos(x))+Csin(x)cos(x)dx=1cos(x)sin(x)dx=1wdw=ln|w|+C=ln|cos(x)|+C

Final step

Hence, by subtracting the two integrals, one gets

int cos^2(x)sin(x)-tan^2(x)cot(x) dx = -cos^3(x)/x - (-lnabs(cos(x)))+C = lnabs(cos(x))-cos^3(x)/x+Ccos2(x)sin(x)tan2(x)cot(x)dx=cos3(x)x(ln|cos(x)|)+C=ln|cos(x)|cos3(x)x+C