Evaluate the following integrals: (4x+2) x^2+x+1 dx - Vidyadip

Question

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

✓

Answer

$\int(4\text{x}+2)\sqrt{\text{x}^2+\text{x}+1}\text{ dx}$
$=2\int(2\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}\text{ dx}$
$\text{Let }\text{x}^2+\text{x}+1=\text{t}$
$\Rightarrow(2\text{x}+1)=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow(2\text{x}+1)\text{dx}=\text{dt}$
$\text{Now, }2\int(2\text{x}+1)\sqrt{\text{x}^2+\text{x}+1}\text{ dx}$
$=2\int\sqrt{\text{t}}\text{ dt}$
$=2\int\text{t}^\frac{1}{2}\text{dt}$
$=2\bigg[\frac{\text{t}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\bigg]+\text{C}$
$=2\times\frac{2}{3}\text{t}^\frac{3}{2}+\text{C}$
$=\frac{4}{3}\text{t}^\frac{3}{2}+\text{C}$
$=\frac{4}{3}(\text{x}^2+\text{x}+1)^\frac{3}{2}+\text{C}$

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

If $\text{A} = \begin{bmatrix} 1 & 2 & 5 \\ 1 & -1 & -1 \\ 2 & 3 & -1 \end{bmatrix}$ find A^–1 and hence solve the system of equations x + 2y + 5z = 10, x – y – z = – 2 and 2x + 3y – z = – 11.

Find the shortest distance between the lines $\frac{\text{x}-2}{-1}=\frac{\text{y}-5}{2}=\frac{\text{z}-0}{3}$ and $\frac{\text{x}-0}{2}=\frac{\text{y}+5}{-1}=\frac{\text{z}-1}{2}.$

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 $3\vec{\text{a}}-2\vec{\text{b}}+\vec{\text{c}}-2\vec{\text{d}}=\vec0$.

Two cards are drawn simultaneosly from a well shuffled deck of 52 cards. Find the probability distribution of the number of the successes, when getting a spade is considered a success.

If A = {a, b, c}, then the relation R = {(b, c)} on A is:

Reflexive only.
Symmetric only.
Transitive only.
Reflexive and transitive only.

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{y}^2=4\text{ax}\text{ at }\Big(\frac{\text{a}}{\text{m}^2},\frac{2\text{a}}{\text{m}}\Big)$

Solve the following L.P.P. graphically:
$\text{Minimise} \text{ }\text{ }\text{ }\text{Z} = 5x + 10\text{y}\\\text{Subect to} \text{ }\text{ }\text{ }\text{ }\text{ }x + \text{2y} \leq 120\\\text{Constraints} \text{ }x + \text{y} \geq 60\\\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }x - \text{2y} \geq 0\\\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{and}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }x, \text{y} \geq 0$

Evaluate the following definite integral as limit of sum:

$\int_\limits{0}^{4}\text{(x}+\text{e}^{2\text{x}})\ \text{dx}$

Find the coordinates of the foot of perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, -1, 3) and C(2, -3, -1).

If $\text{P}\big(\text{x}\big)=\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix},$ then show that P(x)P(y) = P(x + y) = P(y)P(x).