Evaluate the following integrals: 2x^2+3x+4 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\sqrt{2\text{x}^2+3\text{x}+4}\text{dx}$

✓

Answer

$\text{I}=\int\sqrt{2\text{x}^2+3\text{x}+4}\text{dx}$
$=\sqrt2\int\sqrt{\text{x}^2+\frac{3}{2}\text{x}+2}\text{dx}$
$=\sqrt2\int\sqrt{\text{x}^2+\frac{3}{2}\text{x}+\frac{9}{16}+\frac{23}{16}}\text{dx}$
$=\sqrt2\int\sqrt{\Big(\text{x}+\frac{3}{4}\Big)^2+\Big(\frac{\sqrt{23}}{4}\Big)^2}\text{dx}$
$=\sqrt2\begin{Bmatrix}\frac{\big(\text{x}+\frac{3}{4}\big)}{2}\sqrt{\text{x}^2+\frac{3}{2}\text{x}+2}+\frac{23}{32}\\\times\log\bigg|\Big(\text{x}+\frac{3}{4}\Big)+\sqrt{\text{x}^2+\frac{3}{2}\text{x}+2}\bigg|+\text{C}\end{Bmatrix}$
$\therefore\ \text{I}=\frac{4\text{x}+3}{8}\sqrt{2\text{x}^2+3\text{x}+4}+\frac{23\sqrt2}{32}\\\times\log\bigg|\Big(\text{x}+\frac{3}{4}\Big)+\sqrt{\text{x}^2+\frac{3}{2}\text{x}+2}\bigg|+\text{C}$

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 "5 Marks Questions" from Indefinite Integrals→Indefinite 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 angle between the following pairs of lines:$\frac{-\text{x}+2}{-2}=\frac{\text{y}-1}{7}=\frac{\text{z}+3}{-3}$ and $\frac{\text{x}+2}{-1}=\frac{2\text{y}-8}{4}=\frac{\text{z}-5}{4}$

A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. In which of the following cases are the events A and B independent?
A = the card drawn is black,
B = the card drawn is a king.

Evaluate the following integrals:
$\int\frac{\text{dx}}{(\text{x}^2+1)(\text{x}^2+4)}$

Differentiate the following functions from first principles:
$\log\cos\text{x}$

$\text{If y = x}^{x},\text{Prove that } \frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} - \frac{\text{1}}{\text{y}}\bigg(\frac{dy}{dx}\bigg)^{2} - \frac{y}{x} = 0.$

Examine the consistency of the system of equation x + y + z = 1; 2x + 3y + 2z = 2; ax + ay + 2az = 4

Find all point of discontinuity of the function $\text{f(t)}=\frac{1}{\text{t}^2+\text{t}-2},$ where $\text{t}=\frac{1}{\text{x}-1}$

If $\text{y}=\text{e}^{\text{x}}\cos\text{x},$ Prvoe that $\frac{\text{dy}}{\text{dx}}=\sqrt{2}\text{e}^\text{x}.\cos\Big(\text{x}+\frac{\pi}{4}\Big)$

Evaluate the following integrals:
$\int{\frac{\text{e}^{\text{x}}}{\text{e}^{2\text{x}}+5\text{e}^{\text{x}}+6}}\text{dx}$

Minimise Z = x + 2y
subject to $2\text{x}+\text{y}\geq3,\ \text{x}+2\text{y}\geq6,\ \text{x},\ \text{y}\geq0.$
Show that the minimum of Z occurs at more than two points.