Evaluate the following integrals: x^4+1 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{x}\sqrt{\text{x}^4+1}\text{dx}$

✓

Answer

$\text{I}=\int\text{x}\sqrt{\text{x}^4+1}\text{dx}$
$=\int\text{x}\sqrt{(\text{x}^2)^2+1}\text{dx}$
Putting $\text{x}^2=\text{t}$
$\Rightarrow2\text{x dx}=\text{dt}$
$\Rightarrow\text{x dx}=\frac{\text{dt}}{2}$
$\therefore\ \text{I}=\frac{1}{2}\int\sqrt{\text{t}^2+1}\text{dt}$
$=\frac{1}{2}\int\sqrt{\text{t}^2+1}\text{dt}$
$=\frac{1}{2}\Big[\frac{\text{t}}{2}\sqrt{\text{t}^2+1}+\frac{1^2}{2}\log\big|\text{t}+\sqrt{\text{t}^2+1}\big|\Big]+\text{C}$
$=\frac{1}{2}\Big[\frac{\text{x}^2}{2}\sqrt{\text{x}^4+1}+\frac{1}{2}\log\big|\text{x}^2+\sqrt{\text{x}^4+1}\big|\Big]+\text{C}$
$=\frac{\text{x}^2}{4}\sqrt{\text{x}^4+1}+\frac{1}{4}\log\big|\text{x}^2+\sqrt{\text{x}^4+1}+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

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.
$f(x) = x(x + 4)^2$ on $[0,4]$

Evaluate the following integrals:
$\int\frac{\text{x}^2}{(\text{x}^2+1)(3\text{x}^2+4)}\ \text{dx}$

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

Differentiate the following functions with respect to x:
$\sin(\text{x}^\text{x})$

The two adjecent sides of a parallelogram are $2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$ and $\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}}.$ Find the unit vector parallel to one of its diagonals. Also, find its area.

If the tangent to the curve $y = x^3 + ax + b$ at $(1, − 6)$ is parallel to the line $x − y + 5 = 0$, find a and b.

Maximize : z = 9x + 13y subject to 2x + 3y ≤ 18, 2x + y ≤ 10, x ≥ 0, y ≥ 0.

A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. In which of the following cases are the events A and B independent?
A = the card drawn is black,
B = the card drawn is a king.

Find the distance of the point (1, -2, 3) from the plane x - y + z = 5 measured along a line parallel to $\frac{\text{x}}{2}=\frac{\text{y}}{3}=\frac{\text{z}}{-6}.$

Find the equation of the plane that is perpendicular to the plane $5x + 3y + 6z + 8 = 0$ and which contains the line of intersection of the planes $x + 2y + 3z - 4 = 0, 2x + y - z + 5 = 0$.