Question

Integrals

Answer 1

\[\int\limits \sin ^{2}x =\]

Answer 2

\[ \sin ^{2}(x)=(1-\cos(2x))/2\]

Answer 3

\[\int\limits \sqrt{1+\sin2x}\]

Answer 4

integral sin^2(x) dx
Write sin^2(x) as 1/2-1/2 cos(2 x):
= integral (1/2-1/2 cos(2 x)) dx
Integrate the sum term by term and factor out constants:
= integral 1/2 dx-1/2 integral cos(2 x) dx
For the integrand cos(2 x), substitute u = 2 x and du = 2 dx:
= integral 1/2 dx-1/4 integral cos(u) du
The integral of cos(u) is sin(u):
= integral 1/2 dx-(sin(u))/4
The integral of 1/2 is x/2:
= x/2-(sin(u))/4+constant
Substitute back for u = 2 x:
= x/2-1/4 sin(2 x)+constant
Which is equal to:
= 1/2 (x-sin(x) cos(x))+constant