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

Question

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

✓

Answer

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

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 "2 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 Cartesian equation of the following plane:

$\vec{\text{r}}.\Big[(\text{s - 2t})\hat{\text{i}}+(3-\text{t})\hat{\text{j}}+(2\text{s + t})\hat{\text{k}}\Big]=15$

Find the value of k so that the lines x = -y = kz and x - 2 = 2y + 1 = -z + 1 are perpendicular to each other.

Write the function in the simplest form: ${\tan ^{ - 1}}\sqrt {\frac{{1 - \cos x}}{{1 + \cos x}}} ,\;0<x < \pi $

If $\text{P}=\begin{bmatrix}\text{x}&0&0\\0&\text{y}&0\\0&0&\text{z}\end{bmatrix}$ and $\text{Q}=\begin{bmatrix}\text{a}&0&0\\0&\text{b}&0\\0&0&\text{c}\end{bmatrix},$ then prove that $\text{PQ}=\begin{bmatrix}\text{xa}&0&0\\0&\text{yb}&0\\0&0&\text{zc}\end{bmatrix}=\text{QP}.$

If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.

Write the angle between two vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ with magnitudes $\sqrt{3}$ and 2 respectively if $\vec{\text{a}}.\vec{\text{b}}=\sqrt{6}.$

If $\sin^{-1}\Big(\frac{1}{3}\Big)+\cos^{-1}\text{x}=\frac{\pi}{2},$ find x.

What is the angle between vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ with magnitudes 2 and $\sqrt{3}$ respectively? Given $\vec{\text{a}}.\vec{\text{b}}=\sqrt{3}.$

Find fog (2) and gof (1) when : f : R → R; f(x) = x² + 8 and g : R → R; g(x) = 3x³ + 1.

Classify the following functions as injection, surjection or bijection:
f : N → N given by f(x) = x²