Question

integrate:

Answer 1

\[\int\limits_{}^{}\frac{ \sin^2x - 4sinxcosx + 3 \cos^2x }{ sinx + cosx } dx\]

Answer 2

HI!!

Answer 3

maybe try \[\int \frac{1+2\cos^2(x)-4\sin(x)\cos(x)}{\sin(x)+\cos(x)}dx\]?

Answer 4

I tried long division

Answer 5

no, that doesn't seem to help

Answer 6

if i'm not mistaken, can the numerator be factored?

Answer 7

there's got to be a way to simplify the integral

Answer 8

I ended up with

sinx - 5 cosx + (8 cos^2x)/(sinx+cosx)

Answer 9

Vocaloid, yeah I fatored it (sinx - 3cosx)(sinx- cosx)

Answer 10

After long division I had to solve this integral

cos^2x/(sinx+cosx)

Answer 11

That 2cos²x in misty's integral can be converted into cos2x +1

Answer 12

i tried it with wolfram, solution is pages long and a big fat mess, but sometimes wolfram doesn't know the easy way

Answer 13

Not pages

Answer 14

3cosx - sinx + 2sqrt(2) ln((sqrt(2)-1)cosx +sinx/(sqrt(2)-1)cosx-sinx)) + c

Answer 15

That's the final result.

Answer 16

maybe try \[\int \frac{2+cos(2x)-2\sin(2x)}{\sin(x)+\cos(x)}dx\]?

Answer 17

@FaiqRaees , Integrating different angles is a disaster SO HARD.

Answer 18

How I went about to solve it, it's a bit convoluted:
\[\int\limits \frac{ \sin^2x-4sinxcosx+3\cos^2x }{ sinx + cosx }dx=\]
\[\int\limits \frac{ (sinx -cosx)(sinx-3cosx) }{ sinx + cosx }dx\]
Using substitution \(u=sinx + cosx \Rightarrow -du =(sinx -cosx) dx \):
\[\int\limits \frac{ (sinx -cosx)(sinx-3cosx) }{ sinx + cosx }dx=\]
\[\int\limits \frac{ -u+4cosx }{ u } du=-\int\limits 1 du +4 \int\limits \frac{ cosx }{ u } du=\]
\[-u +4 \int\limits \frac{cosx}{u} du = -sinx - cosx + 4 \int\limits \frac {cosx(cosx - sinx)}{cosx + sinx}dx\]

Answer 19

Now:
\[cosx(cosx-sinx)=\cos^2x-cosxsinx=1-sinx^2-cosxsinx=1-sinx(sinx+cosx)\]
So:
\[\int\limits \frac{cosx(cosx-sinx)}{cosx+sinx}dx= \int\limits \frac{1-sinx(sinx+cosx)}{cosx+sinx}dx=\]
\[\int\limits \frac 1 {cosx + sinx} dx - \int\limits sinx dx\]
Finally:
\[cosx+sinx=\sqrt 2 \sin(x+\frac {\pi} 2)\]
So now I should have given you everything necessary to piece it all together and fill in the other bits and pieces. :P

Answer 20

@Bobo-i-bo, Great job mate.