By using the properties of definite integrals, evaluate the integ… - Vidyadip

Question

By using the properties of definite integrals, evaluate the integral $\int\limits_0^4 {\left| {x - 1} \right|dx} $

✓

Answer

Let $I = \int\limits_0^4 {\left| {x - 1} \right|dx} $ ...(i)
Here x - 1 = 0
$ \Rightarrow x = 1 \in \left( {0,4} \right)$
$\therefore$ From eq. (i),
$I = \int\limits_0^1 {\left| {x - 1} \right|dx + \int\limits_1^4 {\left| {x - 1} \right|} dx} $
$= - \int\limits_0^1 {\left( {x - 1} \right)dx + \int\limits_1^4 {\left( {x - 1} \right)} dx} $
$ \Rightarrow I = - \left( {\frac{{{x^2}}}{2} - x} \right)_0^1 + \left( {\frac{{{x^2}}}{2} - x} \right)_1^4$
$= - \left\{ {\left( {\frac{1}{2} - 1} \right) - 0} \right\} + \left\{ {\frac{{16}}{2} - 4 - \left( {\frac{1}{2} - 1} \right)} \right\}$

$= \frac{1}{2} + 8 - 4 + \frac{1}{2}$
= 9 - 4
= 5

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "2 Marks Questions" from Integrals→Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the domain of$\sec^{-1}\text{x}-\tan^{-1}\text{x}$

If $\text{x}=\text{f}(\text{t})$ and $\text{y}=\text{g}(\text{t}),$ then write the value of $\frac{\text{d}^2\text{y}}{\text{dx}^2}.$

Find the principal values of the following:
$\tan^{-1}\Big(\frac{1}{\sqrt3}\Big)$

Find the domain of $\text{f(x)}=\cot\text{x}+\cot^{-1}\text{x}$

A discrete random variable $X$ has the probability distribution given below:

$X:$	$0.5$	$1$	$1.5$	$2$
$P(X):$	$k$	$k^2$	$2k^2$	$k$

Determine the mean of the distribution.

Find the integral of the function $\sin^3 x \cos^3 x$

Differentiate the following with respect to x:
$\cos^{-1}(\sin\text{ x})$

Find $\frac{d y}{d x}$, if x = a cos $\theta$, y = a sin $\theta$.

Write the vector equation of a line given by $\frac{\text{x}-5}{3}=\frac{\text{y}+4}{7}=\frac{\text{z}-6}{2}.$

Differentiate the $\cos (\log x + e^x), x > 0$ w.r.t. x.