Evaluate the following integrals: ^ 1_ -1 |2x+1|dx - Vidyadip

Question

Evaluate the following integrals:
$\int^\limits1_{-1}|2\text{x}+1|\text{dx}$

✓

Answer

We know that,
$\int^\limits1_{-1}|2\text{x}+1|\text{dx}$
$=\int^\limits\frac{1}{2}_{-1}-(2\text{x}+1)\text{dx}+\int\limits_{-\frac{1}{2}}^{1}(2\text{x}+1)\text{dx}$
$=-\Big[\frac{2\text{x}^2}{2}+\text{x}\Big]^{-\frac{1}{2}}_{-1}+\Big[\frac{2\text{x}^2}{2}+\text{x}\Big]^{1}_{-\frac{1}{2}}$
$=-\bigg[\Big(\frac{2}{8}-\frac{1}{2}\Big)-\Big(\frac{2}{2}-1\Big)\bigg]+\bigg[\Big(\frac{2}{2}+1\Big)-\Big(\frac{2}{8}-\frac{1}{2}\Big)\bigg]$
$=-\bigg[\Big(\frac{1}{4}-\frac{1}{2}\Big)-(1-1)\bigg]+\bigg[(1+1)+\Big(\frac{1}{4}-\frac{1}{2}\Big)\bigg]$
$=-\Big[-\frac{1}{4}\Big]++\Big[2+\frac{1}{4}\Big]$
$=\frac{1}{4}+2+\frac{1}{4}$
$=2\frac{1}{2}$

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 "4 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

In a bulb factory, machines A, B and C manufacture 60%, 30% and 10% bulbs respectively. 1%, 2% and 3% of the bulbs produced respectively by A, B and C are found to be defective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine A.

Using properties of determinants, prove the following: $ \begin{bmatrix} \text{ x}&\text{x + y }&\text{x} + 2\text{y}\\ \text{x} + 2\text{y} & \text{x}& \text{x + y }\\\text{x + y}&\text{x} + 2\text{y}& \text{x} \end{bmatrix} = 9\text{y}^{2}(\text{x} + \text{y}). $

Show that the following system of linear equation is inconsistent:
4x − 5y − 2z = 2
5x − 4y + 2z = −2
2x + 2y + 8z = −1

Let $\text{f(x)}=\begin{cases}\frac{1-\sin^3\text{x}}{3\cos^2\text{x}},&\text{if }\text{ x}<\frac{\pi}{2}\\\text{a},&\text{if }\text{ x}=\frac{\pi}{2}\\\frac{\text{b}(1-\sin\text{x})}{(\pi-2\text{x})}^2,&\text{x}>\frac{\pi}{2}\end{cases}$ if f(x) is continuous at $\text{x}=\frac{\pi}{2},$ find a and b.

Find the inverse of the following matrices and verify that A^-1 A = I₃.

$\begin{bmatrix}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}$

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $(2\overrightarrow{\text{a}}+\overrightarrow{\text{b}})$ and$(\overrightarrow{\text{a}}-3\overrightarrow{\text{b}})$ respectively, externally in the ratio 1:2. Also, show that P is the mid point of the line segment RQ.

Find $\frac{\text{dy}}{\text{dx}}$
y = e^x + 10^x + x^x

Evaluate :
$
\int_0^{\frac{\pi}{4}}\left(\frac{\sin x+\cos x}{3+\sin 2 x}\right) d x
$

If $\text{y}=\text{x}^3\log\text{x},$ Prove that $\frac{\text{d}^4\text{y}}{\text{dx}^4}=\frac{6}{\text{x}}$

Suppose 10,000 tickets are sold in a lottery each for Rs. 1. First prize is of Rs. 3000 and the second prize is of Rs. 2000. There are three third prizes of Rs. 500 each. If you buy one ticket, what is your expectation.