Integrate the function in Exercise. x^2+3x - Vidyadip

Question

Integrate the function in Exercise.
$\sqrt{\text{x}^2+3\text{x}}$

✓

Answer

$\int\sqrt{\text{x}^2+3\text{x}}\text{dx}=\int\sqrt{\text{x}^2+3\text{x}+\Bigg(\frac{3}{2}\Bigg)^2-\Bigg(\frac{3}{2}\Bigg)^2}\text{dx}=\int\sqrt{\Bigg(\text{x}+\frac{3}{2}\Bigg)^2-\Bigg(\frac{3}{2}\Bigg)^2}\text{dx}$
$=\frac{\text{x}+\frac{3}{2}}{2}\sqrt{\Bigg(\text{x}+\frac{3}{2}\Bigg)^2-\Bigg(\frac{3}{2}\Bigg)^2}-\frac{\Bigg(\frac{3}{2}\Bigg)^2}{2}\text{log}\Bigg|\text{x}+\frac{3}{2}+\sqrt{\Bigg(\text{x}+\frac{3}{2}\Bigg)^2-\Bigg(\frac{3}{2}\Bigg)^2}\Bigg|+\text{c}$
$\Bigg[\therefore\int\sqrt{\text{x}^2-\text{a}^2}\text{dx}=\frac{\text{x}}{2}\sqrt{\text{x}^2-\text{a}^2}-\frac{\text{a}^2}{2}\text{log}\Bigg|\text{x}+\sqrt{\text{x}^2-\text{a}^2}\Bigg|\Bigg]$
$=\frac{2\text{x}+3}{4}\sqrt{\text{x}^2+3\text{x}}-\frac{9}{8}\text{log}\Bigg|\text{x}+\frac{3}{2}+\sqrt{\text{x}^2+3\text{x}}\Bigg|+\text{c}$

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 "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

Solve system of linear equations, using matrix method.

2x + y - 3z = 0

Evaluate the following definite integrals:
$\int_{0}^\limits{1}\Big(\text{xe}^{2\text{x}}+\sin\frac{\pi\text{x}}{2}\Big)\text{dx}$

Find the matrix A such that
$\text{A}=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\\11&10&9\end{bmatrix}$

If two sides of a triangle be represented by the vectors $\hat{i}+2 \hat{j}+2 \hat{k}$ and $3 \hat{i}-2 \hat{j}+\hat{k}$, then prove that the area of the triangle is $\frac{5}{2} \sqrt{5}$ square units.

Evaluate the following integrals:

$\int\text{e}^{\text{x}}(\log\text{x}+\frac{1}{2})\text{dx}$

In the following, determine the values of constants involved in the definition so that the given function is continuous:
$\text{f(x)}=\begin{cases}\text{k}(\text{x}^2+3\text{x}),&\text{if }\text{ x}<0\\\cos2\text{x},&\text{if }\text{ x}\geq0\end{cases}$

Differentiate following w.r.t. x:
$\sin^\text{n}\big(\text{ax}^2+\text{bx}+\text{c}\big)$

Solve: $\cos\big(\sin^{-1}\text{x}\big)=\frac{1}{6}$

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\text{x}^5+\text{x}^2-\frac{2}{\text{x}},\text{x}\ne0$

Find the vector from the origin O to the centroid of the triangle whose vertices are (1, -1, 2), (2, 1, 3) and (-1, 2, -1).