Question

What is `int sin^3x+3sin^2x+2sinx-5 dx`?

Answer 1

`int (sin^3x +3sin^2x +2sinx -5) dx = -(cosx(2sin^2x +9sinx +16) + 21x)/6 +C`

Explanation:

Using the linearity of the integral:

`int (sin^3x +3sin^2x +2sinx -5) dx = int sin^3xdx +3int sin^2xdx+2 int sinxdx -5 int dx`

Solve the integrals separately:

`(1)`

`int dx = x +c_1`

`(2)`

`int sinxdx = -cosx + c_2`

`(3)`

`int sin^2x = int (1-cos2x)/2 dx`

`int sin^2x = 1/2 int dx -1/2 int cos2x dx`

`int sin^2x = x/2 -(sin2x)/4 +c_3`

`int sin^2x = x/2 -(sinxcosx)/2 +c_3`

`(4)`

`int sin^3xdx = int (1-cos^2x) sinxdx`

`int sin^3xdx = int sinx dx + int cos^2x d(cosx)`

`int sin^3xdx = -cosx + (cos^3x)/3 +c_4`

Putting the partial solutions together:

`int (sin^3x +3sin^2x +2sinx -5) dx = -cosx + (cos^3x)/3 + (3x)/2 -(3sinxcosx)/2 -2cosx -5x +C`

`int (sin^3x +3sin^2x +2sinx -5) dx = (cos^3x)/3 -(3sinxcosx)/2 -3cosx - (7x)/2 +C`

`int (sin^3x +3sin^2x +2sinx -5) dx = (2cos^3x -9sinxcosx -18cosx - 21x)/6 +C`

`int (sin^3x +3sin^2x +2sinx -5) dx = (cosx(2cos^2x -9sinx -18) - 21x)/6 +C`

`int (sin^3x +3sin^2x +2sinx -5) dx = (cosx(2- 2sin^2x -9sinx -18) - 21x)/6 +C`

`int (sin^3x +3sin^2x +2sinx -5) dx = -(cosx(2sin^2x +9sinx +16) + 21x)/6 +C`

Answer 2

There is an error while posting the question. All terms must be multiplied with `dx`, not only the last as given.

Explanation:

The question becomes

`int\ (sin^3x +3sin^2x +2sinx -5) dx`
`=>I= int\ sin^3x\ dx +3int\ sin^2x\ dx+2 int\ sinx\ dx -5 int\ dx`

`=>I= I_1+I_2+I_3+I_4+C`
where `C` is a constant of integration.

Now `I_1=int\ sin^3x\ dx`
`=>I_1= int\ (1-cos^2x) sinx\ dx`

Let us substitute `u=cosx->du=-sinx\ dx`

`=>I_1= int\ (u^2-1)\ du`
`=>I_1= u^3/3-u`

Reversing the substitution we get

`I_1 = (cos^3x)/3 -cosx`

`I_2=3int\ sin^2x\ dx`
`=>I_2 =3 int\ (1-cos2x)/2\ dx`
`=>I_2 =3[ 1/2 int\ dx -1/2 int\ cos2x dx]`
`=>I_2= 3[x/2 -(sin2x)/4 ]`
`=>I_2= (3x)/2 -3/4(sin2x)`

`I_3=2int\ sinx\ dx = -2cosx`

`I_4=-5int\ dx = -5x`

Therefore

`I=( (cos^3x)/3 -cosx)+((3x)/2 -3/4(sin2x))+( -2cosx)+(-5x)+C`
`I=-(9sin2x-4cos^3x+36cosx+42x)/12+C_1`