x x+4 dx - Vidyadip

Question

$\int\frac{\text{x}}{\sqrt{\text{x}+4}}\text{dx}$

✓

Answer

$\int\frac{\text{x}}{\sqrt{\text{x}+4}}\text{dx}$
$=\int\Big(\frac{\text{x}+4-4}{\sqrt{\text{x}+4}}\Big)\text{dx}$
$=\int\Big(\sqrt{\text{x}+4}-\frac{4}{\sqrt{\text{x}+4}}\Big)\text{dx}$
$=\int(\text{x}+4)^\frac{1}{2}\text{dx}-4\int(\text{x}+4)^{-\frac{1}{2}}\text{dx}$
$=\frac{(\text{x}+4)^{\frac{1}{2}+1}}{\frac{1}{2}+1}-4\frac{[\text{x}+4]^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+\text{C}$
$=\frac{2}{3}(\text{x}+4)^\frac{3}{2}-8(\text{x}+4)^\frac{1}{2}+\text{C}$
$=(\text{x}+4)^\frac{1}{2}\Big[\frac{2}{3}(\text{x}+4)-8\Big]+\text{C}$
$=(\text{x}+4)^\frac{1}{2}\Big[\frac{2\text{x}+8-24}{3}\Big]+\text{C}$
$=(\text{x}+4)^\frac{1}{2}\Big[\frac{2\text{x}-16}{3}\Big]+\text{C}$
$=\frac{2}{3}(\text{x}-8)\sqrt{\text{x}+4}+\text{C}$

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

Evaluate the following integrals:
$\int\frac{1}{\sin\text{x}+\sin2\text{x}}\ \text{dx}$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\text{y}=\sin\text{x}$

Find the absolute maximum and minimum values of a function f given by
$\text{f}(\text{x})=12\text{x}^\frac{4}{3}-6\text{x}^\frac{1}{3},\text{x}\in[-1,1]$

Find inverse, by elementary row operations (if possible), of the following matrices.
$\begin{bmatrix}1&3\\-5&7\end{bmatrix}.$

Solve the following differential equation:
$(\text{x}+2\text{y})\text{dx}-(2\text{x}-\text{y})\text{dy}=0$

Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane.

Evaluate the following integrals:
$\int\text{e}^{\text{x}}\sin^2\text{x }\text{dx}$

The vertices A, B, C of triangle ABC have respectively position vector $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ with respect to given origin O. Show that the point D where the bisector of $\angle{\text{A}}$ meets BC has position Vector $\vec{\text{d}}=\frac{\beta\vec{\text{b}}+\gamma\vec{\text{c}}}{\beta+\gamma}$, where $\beta=\big|\vec{\text{c}}-\vec{\text{a}}\big|$ and, $\gamma=\big|\vec{\text{a}}-\vec{\text{b}}\big|$.

If $\sqrt{\text{y}+\text{x}}+\sqrt{\text{y}-\text{x}}=\text{c},$ show that $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}-\sqrt{\frac{\text{y}^2}{\text{x}^2}-1}$

If $\text{A}=\begin{bmatrix}1&0&-3\\2&1&3\\0&1&1\end{bmatrix},$ then verify A² + A = A(A + I), where I is the identity matrix.