Integrate the function f(x) where f(x)= x^2 +sin(x) + sin^2(x)
Answer: Integral= x3/3+ x/2 - cos(x) -1/4 sin(2x) + C Using the general rule that integrating xn results in x(n+1)/(n+1), x2 integrates to x3/3.sin(x) is integrated to -cos(x)To integrate sin2(x), the double angle formula for cos(2x) is needed. As cos(2x)= cos2(x) - sin2(x) and cos2(x) + sin2(x)=1, cos(2x)= 1 - 2 sin2(x). Therefore, sin2(x)= 1/2 - 1/2 cos(2x). Therefore, the integral of sin2(x) is equal to x/2 - 1/4 sin(2x)- the chain rule can be used in reverse to integrate cos(2x).The question has not specified that the integral should be in between boundaries and therefore is an indefinite integral. A constant therefore should be added.
