Question

Find the value of $\int\limits_0^{10\pi } {\left| {\sin x} \right|} dx$
A. $20$
B. $8$
C. $10$
D. $18$

Answer 1

Hint: Use the property of definite integral and then after dividing the integral into proper limits, solve the integral after putting proper limits.

Formula Used:
Separation formula of integration $\int\limits_a^b {f(x)dx = \int\limits_a^x {f(x)dx + \int\limits_x^b {f(x)dx} } } $

Complete step by step solution:
We have the given integral is:
$I = \int\limits_0^{10\pi } {\left| {\sin x} \right|} dx$
We use the formula $\int\limits_a^b {f(x)dx = \int\limits_a^x {f(x)dx + \int\limits_x^b {f(x)dx} } } $
$ = 5\pi \left[ {\int\limits_0^\pi {\sin xdx - \int\limits_\pi ^{2\pi } {\sin xdx} } } \right]$
$ = 5\left[ {\left( { - \left[ {\cos x} \right]_0^\pi } \right) + \left( {\cos x} \right)_\pi ^{2\pi }} \right]$
Substitute the upper limit and lower limit and we get
$ = 5\left[ { - \left( {\cos \pi - \cos 0} \right) + \left( {\cos 2\pi - \cos \pi } \right)} \right]$
$ = 5\left[ { - \left( { - 1 - 1} \right) + \left( {1 - \left( { - 1} \right)} \right)} \right]$
Simplifying and we get
$ = 5 \times 4$
Multiplying and we get
$ = 20$

Option ‘A’ is correct

Note: If and only if an integral has upper and lower bounds, it is said to be definite. There are numerous definite integral formulas and properties that are often utilized in mathematics. You must determine the difference between the values of the integral at the independent variable's defined upper and lower limits in order to determine the value of a definite integral, which is represented as:
$\int\limits_a^b {f(x)dx} $