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 "4 Marks" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the particular solution of the differential equation $\frac{\text{dy}}{\text{dx}}=-4\text{xy}^2$ given that $\text{y}=1.$ when $\text{x}=0.$

Differentiate the following functions with respect to x:
$\log(\text{cosec x}-\cot\text{x})$

If $\text{xy}\log(\text{x}+\text{y})=1,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}(\text{x}^2\text{y}+\text{x}+\text{y})}{\text{x}(\text{xy}^2+\text{x}+\text{y})}$

Show that the four points A, B, C and D with the position vectors $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ and $\vec{\text{d}}$ respectively are coplanar if and only if $3\vec{\text{a}}-2\vec{\text{b}}+\vec{\text{c}}-2\vec{\text{d}}=\vec0$.

The bag A contains 8 white and 7 black balls while the bag B contains 5 white and 4 black balls. One ball is randomly picked up from the bag A and mixed up with the balls in bag B. Then a ball is randomly drawn out from it. Find the probability that ball drawn is white.

Find the point on the curve y = 3x² + 4 at which the tangent is perpendicular to the line whose slop is $-\frac{1}{6}$

An isosceles triangle of vertical angle 2θ is inscribed in a circle of radius a. Show that the area of the triangle is maximum when $\theta = \frac{\pi}{6}.$

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

If $\text{y}=\sin^{-1}\Big(\frac{\text{x}}{1+\text{x}^2}\Big)+\cos^{-1}\Big(\frac{1}{\sqrt{1+\text{x}^2}}\Big), 0<\text{x}<\infty$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{4}{1+\text{x}^2}$

If $\text{A} = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix} \text{and B = } \text{A} = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix}$ are two square matrices, find AB and hence solve the system of linear
equations x – y = 3, 2x + 3y + 4z = 17 and y + 2z = 7.