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Integral of sin⁡(x)2\sin\left(x\right)^{2}

The integral of sin⁡(x)2\sin\left(x\right)^{2} is x2−sin⁡(2x)4+C\frac{x}{2} - \frac{\sin\left(2 x\right)}{4} + C.

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Graph of f(x) = sin(x)^2 with the area from 0 to π shaded-1123450.20.40.60.81f(x)
Graph of f(x) = sin(x)^2 with the area from 0 to π shaded. The shaded area equals the definite integral, about 1.5707963.

How to work it out

  1. 1

    We want an antiderivative of f(x) = sin(x)^2.

    ∫sin⁡(x)2 dx\int \sin\left(x\right)^{2}\,dx
  2. 2

    Method: power-reduction identity sin²x = (1 − cos 2x)/2.

  3. 3

    That gives the antiderivative; add the constant C because any constant disappears when you differentiate:

    ∫sin⁡(x)2 dx=x2−sin⁡(2x)4+C\int \sin\left(x\right)^{2}\,dx = \frac{x}{2} - \frac{\sin\left(2 x\right)}{4} + C
  4. 4

    Check by differentiating the answer; you must get the original function back:

    ddx[x2−sin⁡(2x)4]=0.5−8cos⁡(2x)16\frac{d}{dx}\left[\frac{x}{2} - \frac{\sin\left(2 x\right)}{4}\right] = 0.5 - \frac{8 \cos\left(2 x\right)}{16}
  5. 5

    As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and π:

    ∫0πsin⁡(x)2 dx=[x2−sin⁡(2x)4]0π≈1.5707963\int_{0}^{\pi} \sin\left(x\right)^{2}\,dx = \Big[\frac{x}{2} - \frac{\sin\left(2 x\right)}{4}\Big]_{0}^{\pi} \approx 1.5707963
LaTeX source of the formulas on this page
\[ \int \sin\left(x\right)^{2}\,dx = \frac{x}{2} - \frac{\sin\left(2 x\right)}{4} + C \]
\[ \int \sin\left(x\right)^{2}\,dx \]
\[ \frac{d}{dx}\left[\frac{x}{2} - \frac{\sin\left(2 x\right)}{4}\right] = 0.5 - \frac{8 \cos\left(2 x\right)}{16} \]
\[ \int_{0}^{\pi} \sin\left(x\right)^{2}\,dx = \Big[\frac{x}{2} - \frac{\sin\left(2 x\right)}{4}\Big]_{0}^{\pi} \approx 1.5707963 \]

Definite integrals of the same function

IntervalValue
from 0 to π1.5707963
from 0 to π/20.78539816

Questions

What is the integral of sin(x)^2?

The integral of sin⁡(x)2\sin\left(x\right)^{2} is x2−sin⁡(2x)4+C\frac{x}{2} - \frac{\sin\left(2 x\right)}{4} + C.

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of sin(x)^2 from 0 to π?

About 1.5707963.

More integrals

    Integral of sin(x)^2: x / 2 - sin(2x) / 4 + C (with steps) | WordGPT