Evaluate the following integrals: ^8_2|x-5|dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^8_2|\text{x}-5|\text{dx}$

✓

Answer

$\int\limits^8_2|\text{x}-5|\text{dx}$
We know that,
$|\text{x}-5|=\begin{cases}-(\text{x}-5),&2\leq\text{x}\leq5\\\text{x}-5,&5<\text{x}\leq8\end{cases}$
$\therefore\ \text{I}=\int\limits^8_2|\text{x}-5|\text{dx}$
$\Rightarrow\text{I}=\int\limits^5_2-(\text{x}-5)\text{dx}+\int\limits^8_5(\text{x}-5)\text{dx}$
$\Rightarrow\text{I}=-\Big[\frac{\text{x}^2}{2}-5\text{x}\big]^5_2+\Big[\frac{}{}\frac{\text{x}^2}{2}-5\text{x}\Big]^8_5$
$\Rightarrow\text{I}=\frac{-25}{2}+25+2-10+32-40-\frac{25}{2}+25$
$\Rightarrow\text{I}=9$

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 "3 Marks" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Prove that:
$\cot^{-1} \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} = \frac{x}{2}, 0 < x < \frac{\pi}{2}$

Dot product of a vector with vectore $\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}},2\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}$ and $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ are respectively 4, 0 and 2. Find the vector.

Evaluate the following integrals:
$\int_{0}^\limits{1}\text{xe}^{\text{x}^2}\text{ dx}$

$\text{If}\ \vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ \vec{\text{b}}=2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},$ find a unit vector parallel to the vector $2\vec{\text{a}}-\vec{\text{b}}+3\vec{\text{c}}.$

Integrate the rational function in exercise:
$\frac{\cos\text{x}}{(1-\sin\text{x})(2-\sin\text{x})}$
[Hint: Put sin x = t]

Evaluate the definite integral in Exercise:
$\int\limits^{\frac{\pi}{4}}_0\frac{\sin\text{x}\cos\text{x}}{\cos^{4}\text{x}+\sin^{4}\text{x}}\text{dx}$

Evaluate the following definite integrals:
$\int_{-1}^\limits{1}\frac{1}{\text{x}^2+2\text{x}+5}\text{ dx}$

For each of the differential equations given in find the general solution:
$(\text{x}+3\text{y}^2)\frac{\text{dy}}{\text{dx}}=\text{y}(\text{y}>0)$

Prove that the function f : R → R defined by f (x) = 2x + 5 is one-one.

Find the angle between the pair of lines
$\vec r = (3\hat i + \hat j - 2\hat k) + \lambda (\hat i - \hat j -2\hat k)$ and $\vec r = (2\hat i - \hat j - 56\hat k) + \mu (3\hat i - 5\hat j - 4\hat k)$