Question
What is the definite integral of sine and cosine functions?
Answers
GPT-4o mini
The definite integrals of the sine and cosine functions are calculated over a specific interval. Here are the basic results:
1. **Definite Integral of Sine Function**:
\[
\int_a^b \sin(x) \, dx = -\cos(x) \bigg|_a^b = -\cos(b) + \cos(a) = \cos(a) - \cos(b)
\]
2. **Definite Integral of Cosine Function**:
\[
\int_a^b \cos(x) \, dx = \sin(x) \bigg|_a^b = \sin(b) - \sin(a)
\]
These integrals can be evaluated for any limits \(a\) and \(b\). You can plug in specific values to calculate the definite integral over a specific interval.
1. **Definite Integral of Sine Function**:
\[
\int_a^b \sin(x) \, dx = -\cos(x) \bigg|_a^b = -\cos(b) + \cos(a) = \cos(a) - \cos(b)
\]
2. **Definite Integral of Cosine Function**:
\[
\int_a^b \cos(x) \, dx = \sin(x) \bigg|_a^b = \sin(b) - \sin(a)
\]
These integrals can be evaluated for any limits \(a\) and \(b\). You can plug in specific values to calculate the definite integral over a specific interval.