Evaluate the following integrals: x^2(x^4+4) x^2+4 dx - Vidyadip

Question

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

✓

Answer

Let $\text{I}=\int\frac{\text{x}^2(\text{x}^4+4)}{\text{x}^2+4}\text{ dx}$
$=\int\frac{\text{x}^6+4\text{x}^2}{(\text{x}^2+4)}\text{ dx}$
$=\int\Big[\text{x}^4-4\text{x}^2+20-\frac{80}{\text{x}^2+4}\Big]\text{dx}$
$\text{I}=\frac{\text{x}^5}{5}-\frac{4\text{x}^3}{3}+20\text{x}-80\int\frac{1}{\text{x}^2+4}\text{ dx}+\text{C}_1\ ....(1)$
Let $\text{I}_1=\int\frac{1}{\text{x}^2+4}\text{ dx}$
$\text{I}_1=\int\frac{1}{\text{x}^2+(2)^2}\text{ dx}$
$\text{I}_1=\frac{1}{2}\tan^{-1}\Big(\frac{\text{x}}{2}\Big)+\text{C}_2\ ....(2)$ $\Big[\text{Since},\int\frac{1}{\text{x}^2+\text{a}^2}\text{ dx}=\frac{1}{\text{a}}\tan^{-1}\Big(\frac{\text{x}}{\text{a}}\Big)+\text{C}\Big]$
Using equation (1) and (2)
$\text{I}=\frac{\text{x}^5}{5}-\frac{4\text{x}^3}{3}+20\text{x}-\frac{80}{2}\tan^{-1}\Big(\frac{\text{x}}{2}\Big)+\text{C}$
$\text{I}=\frac{\text{x}^5}{5}-\frac{4\text{x}^3}{3}+20\text{x}-40\tan^{-1}\Big(\frac{\text{x}}{2}\Big)+\text{C}$

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 "5 Marks Questions" 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

Prove that:
$\begin{vmatrix}-\text{bc}&\text{b}^2+\text{bc}&\text{c}^2+\text{bc}\\\text{a}^2+\text{ac}&-\text{ac}&\text{c}^2+\text{ac}\\\text{a}^2+\text{ab}&\text{b}^2+\text{ab}&-\text{ab}\end{vmatrix}$
$=(\text{ab}+\text{bc}+\text{ca})^3$

A company produces two types of goods, A and B, that require gold and silver. Each unit of type A requires 3gm of silver and 1 gm of gold while that of type B requires 1 gm of silver and 2gm of gold. The company can produce 9gm of silver and 8gm of gold. If each unit of type A brings a profit of Rs. 40 and that of type B Rs. 50, find the number of units of each type that the company should produce to maximize the profit. What is the maximum profit?

Show that the differential equation $x \cos \left( \frac { y } { x } \right) \frac { d y } { d x } = y \cos \left( \frac { y } { x } \right) + x$ is homogeneous and solve it.

Evaluate the following intregals:
$\int\frac{3\text{x}+1}{\sqrt{5-2\text{x}-\text{x}^2}}\text{ dx}$

Write the value of $\tan^{-1}\text{x}+\tan^{-1}\Big(\frac{1}{\text{x}}\Big)$ x < 0.

An airline agrees to charter planes for a group. The group needs at least 160 first class seats and at least 300 tourist class seats. The airline must use at least two of its model 314 planes which have 20 first class and 30 tourist class seats. The airline will also use some of its model 535 planes which have 20 first class seats and 60 tourist class seats. Each flight of a model 314 plane costs the company Rs 100,000 and each flight of a model 535 plane costs Rs 150,000. How many of each type of plane should be used to minimize the flight cost? Formulate this as a LPP.

Prove that $\int\limits_{0}^{\text{a}}\text{f(x)}\text{dx}=\int\limits_{0}^{\text{a}}\text{f}(\text{a}-\text{x})\text{dx},$ hence evaluate $\int\limits_{0}^\pi\frac{\text{x}\sin\text{x}}{1+\cos^2\text{x}}\text{dx}.$

If $y^3+3 a x^2+x^3=0$ then prove that:$
\frac{d^2 y}{d x^2}+\frac{2 a^2 x^2}{y^5}=0
$

Show that the curves $2x = y^2$ and $2xy = k$ cut at right angles, if $k^2 = 8.$

Solve the following systems of linear equations by cramer's rule:

2x - 3y - 4z = 29,

-2x + 5y - z = -15,

3x - y + 5z = -11