Evaluate the following integrals: x^2+9 x^4+81 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\text{x}^2+9}{\text{x}^4+81}\ \text{dx}$

✓

Answer

Let $\text{I}=\int\frac{\text{x}^2+9}{\text{x}^4+81}\ \text{dx}$

Dividing numerator and denominator by x²

$\text{I}=\int\frac{1+\frac{9}{\text{x}^2}}{\text{x}^2+\frac{81}{\text{x}^2}}\ \text{dx}$

$=\int\frac{1+\frac{9}{\text{x}^2}}{\Big(\text{x}-\frac{9}{\text{x}}\Big)^2+18}$

Let $\Big(\text{x}-\frac{9}{\text{x}}\Big)=\text{t}\Rightarrow\Big(1+\frac{9}{\text{x}^2}\Big)\text{dx}=\text{dt}$

$\therefore\text{I}=\int\frac{\text{dt}}{\text{t}^2+18}$

$\text{I}=\frac{1}{3\sqrt{2}}\tan^{-1}\Big(\frac{\text{t}}{3\sqrt{2}}\Big)+\text{C}$

Thus,

$\text{I}=\frac{1}{3\sqrt{2}}\tan^{-1}\Big(\frac{\text{x}^2-9}{3\sqrt{2}}\Big)+\text{C}$

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 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

A rubber company is engaged in producing three types of tyres A, B and C. Each type requires processing in two plants, Plant I and Plant II. The capacities of the two plants, in number of tyres per day, are as follows:

Plant	A	B	C
I	50	100	100
II	60	60	200

The monthly demand for tyre A, B and C is 2500, 3000 and 7000 respectively. If plant I costs Rs. 2500 per day, and plant II costs Rs. 3500 per day to operate, how many days should each be run per month to minimize cost while meeting the demand? Formulate the problem as LPP.

Find the intervals in which the following functions are increasing or decreasing.
f(x) = x³ - 6x² + 9x + 15

Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half - life is 1590 years. What percentage will disappear in one year?

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}\cos(\text{x}+\text{y})&-\sin(\text{x}+\text{y})&\cos2\text{y}\\\sin\text{x}&\cos\text{x}&\sin\text{y}\\-\cos\text{x}&\sin\text{x}&-\cos\text{y} \end{vmatrix}$

Show that $2\tan^{-1}\text{x}+\sin^{-1}\frac{2\text{x}}{1+\text{x}^2}$ is constant for $\text{x}\geq1,$ find that constant.

Using definite intergeals, find the area of the circle x²+ y² = a².

Evaluate: $\int\limits^{\pi}_{0} \frac{\text{x}}{1 + \sin \alpha \sin x} \text{dx}.$

Differentiate the following functions with respect to x:
$\tan^{-1}\bigg[\frac{\text{x}^\frac{1}{3}+\text{a}^{\frac{1}{3}}}{1-(\text{ax})^\frac{1}{3}}\bigg]$

For each of the differential equations given in find the general solution:
$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}-\text{x}+\text{xy}\cot\text{x}=0\ (\text{x}\neq0)$

A fair coin is tossed four times. Let X denote the number of heads occuring. Find the probability distribution, mean and variance of X.