Question

How do you find the definite integral for \[\cos \left( x \right)\] in the interval \[\left[ { - \pi ,\pi } \right]\]?

Answer 1

Hint: In the given question, we have been given a trigonometric function. We have to integrate it. To do that, we make the substitution of the integral using the standard results. Then we make substitutions and use the standard integration result, apply the formula of the definite integral using the limits to calculate the answer.

Formula Used:
We are going to use the formula of integration of \[\cos \left( x \right)\], which is,
\[\int {\cos \left( x \right)dx} = \sin \left( x \right)\]

Complete step by step answer:
We have to calculate the integral of \[\cos \left( x \right)\].
\[I = \int {\cos \left( x \right)dx} \]
We know,
\[\dfrac{{d\left( {\sin x + c} \right)}}{{dx}} = \cos x\]
Hence, we have,
\[\int {\cos \left( x \right)dx} = \sin x + c\]
Now we put in the value of the limits,

\[\int\limits_{ - \pi }^\pi {\cos \left( x \right)dx} = {\left[ {\sin x} \right]^{ + \pi }}_{ - \pi } = \sin \left( \pi \right) - \sin \left( { - \pi } \right) = \sin \pi + \sin \pi = 0\]

Additional Information:
Integration is the opposite of differentiation. In differentiation, we “break” things for examining how they behave separately. While, in integration, we combine the expressions so as to see their collective behavior. If we have a definite integral, then we calculate its value by putting in the upper limit into the result, then putting in the lower limit into the result, and then subtracting the two. A definite integral is the one which looks like,
\[\int\limits_b^a {some{\rm{ }}\exp ression} \].

Note:
In the given question, we had to find the integral of cosine. We did that by using the standard result to evaluate the value of the expression. Then we put in the values of the limits and evaluated the answer.