Integrate the function in Exercise: 1 x^ 2 (x^ 2 +1)^ 3 4 - Vidyadip

Question

Integrate the function in Exercise:
$\frac{1}{\text{x}^{2}(\text{x}^{2}+1)^{\frac{3}{4}}}$

✓

Answer

$\frac{1}{\text{x}^{2}(\text{x}^{2}+1)^{\frac{3}{4}}}$Multipling and dividing by $\text{x}^{-3}$, we obtain
$\frac{\text{x}^{-3}}{\text{x}^{2}.\text{x}^{-3}(\text{x}^{4}+1)^{\frac{3}{4}}}=\frac{\text{x}^{-3}(\text{x}^{4}+1)^{\frac{3}{2}}}{\text{x}^{2}.\text{x}^{-3}}$
$=\frac{(\text{x}^{4}+1)^{\frac{-3}{4}}}{\text{x}^{5}.(\text{x)}^{4^{-\frac{3}{4}}}}$
$=\frac{1}{\text{x}^{5}}\bigg(\frac{\text{x}^{4}+1}{\text{x}^{4}}\bigg)^{-\frac{3}{4}}$
$\text{Let}\frac{1}{\text{x}^{4}}=\text{t} \Rightarrow-\frac{4}{\text{x}^{5}}\text{dx}=\text{dt}\Rightarrow\frac{1}{\text{x}^{5}}\text{dx}=-\frac{\text{dt}}{4}$
$\therefore\int\frac{1}{\text{x}^{2}(\text{x}^{4}+1)^{\frac{3}{4}}}\text{dx}=\int\frac{1}{\text{x}^{5}}\bigg(1+\frac{1}{\text{x}^{4}}\bigg)^{-\frac{3}{4}}\text{dx}$
$=-\frac{1}{4}\int(1+\text{t)}^{-\frac{3}{4}}\text{dt}$
$=-\frac{1}{4}\Bigg[\frac{(1+\text{t)}^{-\frac{3}{4}}}{\frac{1}{4}}\Bigg]+\text{C}$
$=-\frac{1}{4}\frac{\bigg(1+\frac{1}{\text{x}^{4}}\Bigg)^{\frac{1}{4}}}{\frac{1}{4}}+\text{C}$
$=-\Bigg(1+\frac{1}{\text{x}^{4}}\Bigg)^{\frac{1}{4}}+\text{C}$

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 "5 Marks Questions" from INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{x}^{\frac{2}{3}}+\text{y}^{\frac{2}{3}}=2\text{ at }(1,1)$

Verify Lagrange's mean value theorem for the following function on the indicated intervals. Find a point $'c\ '$ in the indicated interval as stated by the Lagrange's mean value theorem. $f(x) = x^2- 1$ on $[2, 3]$

Differentiate the following functions with respect to x:
$\tan^{-1}\Big\{\frac{\text{x}}{\sqrt{\text{a}^2-\text{x}^2}}\Big\},-\text{a}<\text{x}<\text{a}$

Evaluate the following integrals:$\int(\log\text{x})^2\cdot\text{x dx}$

Evaluate: $\int\limits^\pi_0\frac{x\tan x}{\text{sec }x.\text{ cosec }x}\text{ d}x.$

For the matrix $\text{A} = \begin{bmatrix}3&2\\1&1\end{bmatrix},$ find the numbers $a$ and $b$ such that $A^2 + aA + bI = 0.$

Evaluate the following integrals:
$\int(\tan^{-1}\text{x}^2)\text{x dx}$

Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x coordinate (abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point.

A merchant plans to sell two types of personal computers a desktop model and a portable model that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs. 70 lakhs and his profit on the desktop model is Rs. 4500 and on the portable model is Rs. 5000. Make an LPP and solve it graphically.

$\text{If} \overrightarrow{a} \times \overrightarrow{b} = \overrightarrow{c} \times \overrightarrow{d} and \overrightarrow{a} \times\overrightarrow{c} = \overrightarrow{b} \times\overrightarrow{d},$ show that $\overrightarrow{a} - \overrightarrow{d}$ is parallel to $\overrightarrow{b}-\overrightarrow{c},$ where $\overrightarrow{a} \neq\overrightarrow{d} and \overrightarrow{b}\neq\overrightarrow{c}.$