Integral of sin(x)cos(x)
The integral of sin(x)cos(x) is 2sin(x)2+C.
Graph of f(x) = sin(x) * cos(x) with the area from 0 to π/2 shaded. The shaded area equals the definite integral, about 0.5.How to work it out
- 1
We want an antiderivative of f(x) = sin(x) * cos(x).
∫sin(x)cos(x)dx - 2
Method: substitution u = sin x.
- 3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
∫sin(x)cos(x)dx=2sin(x)2+C - 4
Check by differentiating the answer; you must get the original function back:
dxd[2sin(x)2]=422sin(x)cos(x) - 5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and π/2:
∫02πsin(x)cos(x)dx=[2sin(x)2]02π≈0.5
LaTeX source of the formulas on this page
\[ \int \sin\left(x\right) \cos\left(x\right)\,dx = \frac{\sin\left(x\right)^{2}}{2} + C \]
\[ \int \sin\left(x\right) \cos\left(x\right)\,dx \]
\[ \frac{d}{dx}\left[\frac{\sin\left(x\right)^{2}}{2}\right] = \frac{2 2 \sin\left(x\right) \cos\left(x\right)}{4} \]
\[ \int_{0}^{\frac{\pi}{2}} \sin\left(x\right) \cos\left(x\right)\,dx = \Big[\frac{\sin\left(x\right)^{2}}{2}\Big]_{0}^{\frac{\pi}{2}} \approx 0.5 \]Definite integrals of the same function
| Interval | Value |
|---|
| from 0 to π/2 | 0.5 |
| from 0 to 0.7853981633974483 | 0.25 |
Questions
What is the integral of sin(x) * cos(x)?
The integral of sin(x)cos(x) is 2sin(x)2+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) * cos(x) from 0 to π/2?
About 0.5.