Evaluate the Integral integral of sin(x)^4 with respect to x

Problem

(∫_^)(sin4(x)*d(x))

Solution

1. Apply the power-reduction formula for sin2(x) to rewrite the integrand.

sin2(x)=(1−cos(2*x))/2

2. Square the expression to represent sin4(x)

sin4(x)=((1−cos(2*x))/2)2

3. Expand the square using the binomial identity (a−b)2=a2−2*a*b+b2

sin4(x)=1/4*(1−2*cos(2*x)+cos2(2*x))

4. Apply the power-reduction formula again for the cos2(2*x) term.

cos2(2*x)=(1+cos(4*x))/2

5. Substitute and simplify the entire expression for the integrand.

sin4(x)=1/4*(1−2*cos(2*x)+(1+cos(4*x))/2)

sin4(x)=1/4*(1−2*cos(2*x)+1/2+cos(4*x)/2)

sin4(x)=3/8−1/2*cos(2*x)+1/8*cos(4*x)

6. Integrate term by term with respect to x

(∫_^)((3/8−1/2*cos(2*x)+1/8*cos(4*x))*d(x))

7. Evaluate the integrals using the rule (∫_^)(cos(a*x)*d(x))=sin(a*x)/a

3/8*x−1/4*sin(2*x)+1/32*sin(4*x)+C

Final Answer

(∫_^)(sin4(x)*d(x))=(3*x)/8−sin(2*x)/4+sin(4*x)/32+C

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