Question

Evaluate: `int dx/(sin^2x+sin^2x*cos^2x+cos^2x)` ?

Answer 1

`int \ 1/(sin^2x+sin^2x*cos^2x+cos^2x) \ dx = (sqrt(5))/5 \ arctan((sqrt(5)tan(2x))/2) + C`

Explanation:

We seek:

`I = int \ 1/(sin^2x+sin^2x*cos^2x+cos^2x) \ dx`

Now, using the identities:

`sin^2A+cos^2A-=1`,
`sin2A-=2sinAcosA`,

We can write:

`I = int \ 1/(1+sin^2x*cos^2x) \ dx`

`\ \ = int \ 1/(1+(sinx \ cosx)^2) \ dx`

`\ \ = int \ 1/(1+(1/2sin2x)^2) \ dx`

`\ \ = int \ 1/(1+1/4sin^2(2x)) \ dx`

`\ \ = 4 \ int \ 1/(4+sin^2(2x)) \ dx`

We can initially perform a trivial substitution to simplify the integral, Let:

`u=2x => (du)/dx=2`

So that:

`I = 4 \ int \ 1/(4+sin^2(u)) \ (1/2) \ du`

`\ \ = 2 \ int \ 1/(4+sin^2(u)) \ du`

Now, for this particular style of integral we employ a standard technique to eliminate trigonometry from the problem. We can write all trigonometry terms in terms of the tangent and secant function and utilizing `1+tan^2A-=sec^2A`. So, we now perform a substitution:

`v = tanu => (dv)/(du) = sec^2u = 1+tan^2u=1+v^2`

And:

`sinx -= sinx/cosx * cos x = tanx /secx`

`:. sin^2x = tanx^2 /sec^2x = v^2/(1+v^2)`

Therefore:

`I = 2 \ int \ 1/(4+sin^2(u)) \ du`

`\ \ = 2 \ int \ 1/(4+v^2/(1+v^2)) \ (1/(1+v^2)) \ dv`

`\ \ = 2 \ int \ 1/((4(1+v^2)+v^2)/(1+v^2)) \ (1/(1+v^2)) \ dv`

`\ \ = 2 \ int \ 1/(4+4v^2+v^2) \ dv`

`\ \ = 2 \ int \ 1/(4+5v^2) \ dv`

We can now perform a final substitution to put this last integral into a standard form, by using:

`w = (sqrt(5)v)/2 => (dw)/(dv) = (sqrt(5))/2`

So that:

`I = 2 \ int \ 1/(4+5((2w)/sqrt(5))^2) \ (2/sqrt(5)) \ dw`

`\ \ = 4/sqrt(5) \ int \ 1/(4+5((4w^2)/5)) \ dw`

`\ \ = 4/sqrt(5) sqrt(5)/sqrt(5) \ int \ 1/(4+4w^2) \ dw`

`\ \ = (sqrt(5))/5 \ int \ 1/(1+w^2) \ dw`

Which is a standard integral, so we can directly integrate to get:

`I = (sqrt(5))/5 \ arctan(w) +C`

Then we restore the substitution `w = (sqrt(5)v)/2`:

`I = (sqrt(5))/5 \ arctan((sqrt(5)v)/2) +C`

Then we restore the substitution `v = tanu`

`I = (sqrt(5))/5 \ arctan((sqrt(5)tanu)/2) + C`

Finally, we restore the substitution `u=2x`

`I = (sqrt(5))/5 \ arctan((sqrt(5)tan(2x))/2) + C`