Find the integral: (x-1 x )^ 2 d x - Vidyadip

Question

Find the integral: $\int\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2} d x$

✓

Answer

$\int\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2} d x$
= $\int\left(x+\frac{1}{x}-2\right) d x$
= $\int x d x+\int \frac{1}{x} d x-2 \int 1 d x$
Now we know that,
$^{\int x^{n} d x}=\frac{x^{n+1}}{n+1}+c$
Therefore,
$\int x d x+\int \frac{1}{x} d x-2 \int 1 d x$ = $\frac{x^{2}}{2}+\log |x|-2 x+C$

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 Questions" 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

Integrate the function $\frac{{{{\sec }^2}x}}{{\sqrt {{{\tan }^2}x + 4} }}$

Compute the products AB and BA whichever exists the following cases:
$\text{A}=\begin{bmatrix}3&2\\-1&0\\-1&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}4&5&6\\0&1&2\end{bmatrix}$

Find the maximum and minimum values, if any, of the function given by:
h(x) = Sin (2x) + 5

Find the principal values:
$\tan^{-1}(-1)$

Suppose X has a binomial distribution with n = 6 and p = $\frac{1}{2}.$ Show that X = 3 is the most likely outcome.

Find the interval where function $f(x)=x^3-3 x$ is decreasing.

Write the cofactor of $a_{12}$ in the following matrix $\begin{bmatrix}2&-3&5\\6&0&4\\1&5&-7\end{bmatrix}$

Let * be a binary operation, on the set of all non-zero real numbers, given by
$\text{a}\times\text{b}=\frac{\text{ab}}{5}\ \forall\text{ a, b}\in\text{R}-\{0\}$
Write the value of x given by 2 * (x * 5) = 10.

For the curve $y = 5x – 2x^3$, if x increases at the rate of $2$ units/sec, then find the rate of change of the slope of the curve when $x = 3$.

$\text{If y}=\begin{vmatrix}\text{f(x)} & \text{g(x)} & \text{h(x)} \\ \text{l} & \text{m} & \text{n} \\ \text{a} & \text{b} & \text{c} \end{vmatrix},\ \text{prove that}\frac{\text{dy}}{\text{dx}}= \begin{vmatrix}\text{f(x)} & \text{g'(x)} & \text{h'(x)} \\ \text{l} & \text{m} & \text{n} \\ \text{a} & \text{b} & \text{c}\end{vmatrix}$