Question

I = cos^2(x) dxइंटीग्रेशन ऑफ 1 ​

Answer 1

Answer:

�

�

�

�

� �

� � �

� ��

� �

sin4

(x) cos2

(x) dx

Compute sin4(x) cos2(x) dx.

Solution

Because all of the exponents in this problem are even, our chosen solution involves half angle formulas:

cos2 θ = 1 + cos(2θ)

2

sin2 θ = 1 − cos(2θ)

. 2

Because we have to do a lot of writing before we actually integrate anything,

we’ll start with some “side work” to convert the integrand into something we

know how to integrate.

sin4 x cos2 x = (sin2 x)

2 cos2 x

� �2 � � 1 − cos(2x) 1 + cos(2x) = 2 2

� � � � 1 − 2 cos(2x) + cos2(2x) 1 + cos(2x) = 4 2

1 − 2 cos(2x) + cos2(2x) + cos(2x) − 2 cos2(2x) + cos3(2x) = 8

1 − cos(2x) − cos2(2x) + cos3(2x) = 8

This is all the side work we need to do here, because we know that:

cos2(2x) = x

+ sin(2x) + c1 and

2 4

cos3(2x) = 1 sin(2x) − 1

sin3(2x) + c2. 2 6

We conclude that:

sin4 x cos2 x dx = 1 − cos(2x) − cos2(2x) + cos3(2x) dx

8

1 1 x sin(2x) = 8

x − 2 sin(2x) − 2 + 4 + c1

+

1 sin(2x) − 1

sin3(2x) + c2 2 6

1 x sin(2x) 1 = 8 2 − 4 − 6

sin3(2x) + C

1

x sin(2x) sin3(2x) = + C 16 − 32 − 48

It’s difficult to check that this is the correct answer. If C = 0 this is an odd

function which is at least consistent with the integrand being an even function.

Step-by-step explanation: