# Integral of trigonometric functions

• darkmagic
In summary, the general formula for finding the integral of a trigonometric function is ∫sin(x)dx = -cos(x) + C or ∫cos(x)dx = sin(x) + C. To integrate trigonometric functions with powers, power reduction formulas are used, along with substitution and integration by parts methods. Trigonometric identities can also be used to simplify and solve integrals. Special cases such as odd or even powers, square roots, and constants may require specific techniques. The process for evaluating definite integrals involves finding the antiderivative, substituting the limits of integration, and then subtracting the values.
darkmagic

## Homework Statement

How can I integrate this:

$$\int sin (nt) sin (n \pi t) dt$$

This actually in the Fourier series.

Use the trig identity:

$$\sin\theta\sin\phi=\frac{\cos{(\theta-\phi)}-\cos{(\theta+\phi)}}{2}$$

You can use the following identity:

$$\sin a \sin b=\frac{\cos(a-b)-\cos(a+b)} 2$$

## 1. What is the general formula for finding the integral of a trigonometric function?

The general formula for finding the integral of a trigonometric function is ∫sin(x)dx = -cos(x) + C or ∫cos(x)dx = sin(x) + C, where C is the constant of integration.

## 2. How do you integrate trigonometric functions with powers?

To integrate trigonometric functions with powers, use the power reduction formulas: sin^n(x) = (1-cos^2(x))^(n/2) and cos^n(x) = (1+cos^2(x))^(n/2). Then use substitution and integration by parts methods to solve the integral.

## 3. Can the integral of trigonometric functions be solved using trigonometric identities?

Yes, trigonometric identities such as the double angle formulas, half angle formulas, and sum and difference formulas can be used to simplify and solve integrals of trigonometric functions.

## 4. Are there any special cases when integrating trigonometric functions?

Yes, when integrating trigonometric functions, special cases such as odd or even powers, the presence of a square root, or the presence of a constant may require the use of specific techniques such as substitution or trigonometric identities.

## 5. What is the process for evaluating definite integrals of trigonometric functions?

The process for evaluating definite integrals of trigonometric functions involves finding the antiderivative of the function, substituting the upper and lower limits of integration, and then subtracting the values to find the final answer.

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