"Write the integral as a sum of integrals..." was answered by our experts

High School
Calculus and Analysis
Calculus 2
Applications of integrals

enseignement20

Open question

2022-08-18

Write the integral as a sum of integrals without absolute values and evaluate.

\int_{0}^{π} | \cos x | d x

Answer & Explanation

klunnalegj6

Beginner2022-08-19Added 7 answers

Step 1
Given-

\int_{0}^{π} | \cos x | d x

To find- Evaluating the above integral using proper identities.
Concept used- As the given integral is in modulus, so break the limit as per the given function.
Step 2
Explanation- As the above integral in modulus, so using the property of modulus, proceeding as follows,
Let,

I = \int_{0}^{π} | \cos x | d x

Now, solving further,
Splitting the function from 0 to

\frac{π}{2}

and

\frac{π}{2}

π

and we know that

\cos x

is postive for 0 to

\frac{π}{2}

and negtaive for

\frac{π}{2}

π

I = \int_{0}^{\frac{π}{2}} \cos x d x - \int_{\frac{π}{2}}^{π} \cos x d x

= {[\sin x]}_{0}^{\frac{π}{2}} - {[\sin x]}_{\frac{π}{2}}^{π}

= [\sin \frac{π}{2} - \sin 0] - [\sin π - \sin \frac{π}{2}]

=1-0-0+1
=2
Answer- The value of the integral

\int_{0}^{π} | \cos x | d x

is 2.

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