The solution to the integral of sin^2(x) requires you to recall principles of both trigonometry and calculus. Don't conclude that since the integral of sin(x) equals -cos(x), the integral of sin^2(x) should equal -cos^2(x); in fact, the answer does not contain a cosine at all. You cannot directly integrate sin^2(x). Use trigonometric identities and calculus substitution rules to solve the problem.
Use the half angle formula, sin^2(x) = 1/2*(1 - cos(2x)) and substitute into the integral so it becomes 1/2 times the integral of (1 - cos(2x)) dx.
Set u = 2x and du = 2dx to perform u substitution on the integral. Since dx = du/2, the result is 1/4 times the integral of (1 - cos(u)) du.
Integrate the equation. Since the integral of 1du is u, and the integral of cos(u) du is sin(u), the result is 1/4*(u - sin(u)) + c.
Substitute u back into the equation to get 1/4*(2x - sin(2x)) + c. Simplify to get x/2 - (sin(x))/4 + c.
Style Your World With Color
Let your clothes speak for themselves with this powerhouse hue.View Article
Create balance and growth throughout your wardrobe.View Article
See if her signature black pairs well with your personal style.View Article
Explore a range of deep greens with the year's "it" colors.View Article
- For a definite integral, eliminate the constant in the answer and evaluate the answer over the interval specified in the problem. If the interval is 0 to 1, for example, evaluate [1/2 - sin(1)/4] - [0/2 - sin(0)/4)].