Integral of
The integral of is .
How to work it out
- 1
We want an antiderivative of f(x) = sin(3x).
- 2
Method: ∫sin(kx) dx = −cos(kx)/k.
- 3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
- 4
Check by differentiating the answer; you must get the original function back:
- 5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and π:
LaTeX source of the formulas on this page
\[ \int \sin\left(3 x\right)\,dx = \frac{-\cos\left(3 x\right)}{3} + C \]
\[ \int \sin\left(3 x\right)\,dx \]
\[ \frac{d}{dx}\left[\frac{-\cos\left(3 x\right)}{3}\right] = \frac{9 \sin\left(3 x\right)}{9} \]
\[ \int_{0}^{\pi} \sin\left(3 x\right)\,dx = \Big[\frac{-\cos\left(3 x\right)}{3}\Big]_{0}^{\pi} \approx 0.66666667 \]Definite integrals of the same function
| Interval | Value |
|---|---|
| from 0 to π | 0.66666667 |
| from 0 to π/2 | 0.33333333 |
Questions
What is the integral of sin(3x)?
The integral of is .
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(3x) from 0 to π?
About 0.66666667.