Integrate: int(3sin(x)+2cos(x))(2sin(x)+3cos(x))dx

Michael Maggard

Michael Maggard

Answered question

2022-01-07

Integrate: 3sin(x)+2cos(x)2sin(x)+3cos(x)dx

Answer & Explanation

Jillian Edgerton

Jillian Edgerton

Beginner2022-01-08Added 34 answers

We can split this into two integrals:
3sin(x)2sin(x)+3cos(x)dx+2cos(x)2sin(x)+3cos(x)dx
Focusing on the second integral we find:
cos(x)2sin(x)+3cos(x)dx=12tan(x)+3dx
Make the substitution u=2tan(x)+3 which makes
du=2sec2(x)dx=2(tan2(x)+1)dx=2(u2+1)dx
Thus we have
cos(x)2sin(x)+3cos(x)dx=121u(u2+1)du=12(1uuu2+1)du
This integral can be computed by another substitution:
cos(x)2sin(x)+3cos(x)dx=12ln|2tan(x)+3|12ln|(2tan(x)+3)2+1|+C
That completes the second integral. The first can be handled in a similar manner.
veiga34

veiga34

Beginner2022-01-09Added 32 answers

It is a pity you did not have a minus-sign in the numerator, since
Dln(3cosx+2sinx)=2cosx3sinx3cosx+2sinx
but let us see how we can use this fact anyways.
Let us aim at writing
2cosx+3sinx3cosx+2sinx=c12cosx3sinx3cosx+2sinx+c23cosx+2sinx3cosx+2sinx
since both those terms are easy to integrate. This leads us to the linear equations 2=2c1+3c2 and 3=3c1+2c2. The solution to this system is c1=513 and c2=1213. Thus
2cosx+3sinx3cosx+2sinxdx=5132cosx3sinx3cosx+2sinxdx+12133cosx+2sinx3cosx+2sinxdx.
I guess you can take it from here?
karton

karton

Expert2022-01-11Added 613 answers

Let there be a right triangle with angle A(not with measurement 90 deg) whose adjacent has measurement 3 and whose opposite has measerement 2. Therefore the hypotenuse of the triangle has measurement 13
So this means

sin(A)=213 and cos(A)=313.3sin(x)+2cos(X)=13(313sin(x)+213cos(x))=13(cos(A)sin(x)+sin(A)cos(x))=13cos(xA)3sin(x)+2cos(x)2sin(x)+3cos(x)dx=13sin(x+A)13cos(xA)dx=sin(xA+2A)cos(xA)dx=sin(xA)cos(2A)+sin(2A)cos(xa)cos(xA)=cos(2A)sin(xA)cos(xA)dx+sin(2A)cos(xA)cos(xA)dx=(cos2(A)sin2(A))sin(xA)cos(xA)dx+2sin(A)cos(A)1dx=((313)2(213)2)duu+2213313x+C(note: where u=cos(xA) and so du=sin(xA)dx)=(913413)(ln|u|)+1213x+C=513ln|cos(xA)|+1213x+C=513ln|cos(x)cos(A)+sin(x)sin(A)|+1213x+C=513ln|113|513ln|3cos(x)+2sin(x)|+1213x+C=513ln|3cos(x)+2sin(x)|+1213x513ln|113|+C=513ln|3cos(x)+2sin(x)|+1213x+K

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