Question

What is the integral of ∫[sin(2⋅π⋅x)]2?

Answer 1

Answer:

∫

sin

2

(

2

π

x

)

d

x

=

x

2

−

1

8

π

sin

(

4

π

x

)

+

C

Explanation:

Use the trigonometric identity:

sin

2

θ

=

1

−

cos

2

θ

2

We have:

∫

sin

2

(

2

π

x

)

d

x

=

∫

1

−

cos

4

π

x

2

d

x

using linearity:

∫

sin

2

(

2

π

x

)

d

x

=

1

2

∫

d

x

−

1

2

∫

cos

4

π

x

d

x

=

x

2

−

1

8

π

∫

cos

(

4

π

x

)

d

(

4

π

x

)

=

x

2

−

1

8

π

sin

(

4

π

x

)

+

C