Evaluate the following integral: x^4+1 x^2+1 dx - Vidyadip

Question

Evaluate the following integral:
$\int\frac{\text{x}^4+1}{\text{x}^2+1}\text{ dx}$

✓

Answer

$\int\Big(\frac{\text{x}^4+1}{\text{x}^2+1}\Big)\text{ dx}$
$=\int\Big(\frac{\text{x}^4-1+1+1}{\text{x}^2+1}\Big)\text{ dx}$
$=\int\Big[\frac{(\text{x}^4-1)}{\text{x}^2+1}+\frac{2}{\text{x}^2+1}\Big]\text{ dx}$
$=\int\Big[\frac{(\text{x}^2-1)(\text{x}^2+1)}{(\text{x}^2+1)}+\frac{2}{\text{x}^2+1}\Big]\text{ dx}$
$=\int\Big[(\text{x}^2-1)+\frac{2}{\text{x}^2+1}\Big]\text{ dx}$
$=\frac{\text{x}^3}{3}-\text{x}+2\tan^{-1}(\text{x})+\text{C}$

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

Divide 64 into two parts such that the sum of the cubes of two parts is minimum.

Show that the following curves intersect orthogonally at the indicated points:
$y^2 = 8x $and $2x^2 + y^2 = 10 $at $\big(1,2\sqrt{2})$

Find the equation of the tangent line to the curve $y = x^2 - 2x + 7$ which is perpendicular to the line $5y - 15x = 13.$

Find the area of the region common to the parabolas $4y^2= 9x$ and $3x^2= 16y.$

Evaluate the following integrals:$\int\limits^{\text{b}}_{\text{a}}\frac{\text{x}^{\frac{1}{\text{n}}}}{\text{x}^\frac{1}{\text{n}}+\big(\text{a}+\text{b}-\text{x}\big)^{\frac{1}{\text{n}}}}\text{ dx},\text{ n}\in\text{N},\text{n}\leq2$

If $\text{y}=\text{cosec}^{-1}\text{x},\text{x}>1$ prove that $\text{x}(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}+(2\text{x}^2-1)\frac{\text{dy}}{\text{dx}}=0.$

Find the equation of the plane passing through the point (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane x - 2y + 4z = 10.

Solve the following differential equation:$\frac{\text{dy}}{\text{dx}}+2\text{y}=6\text{e}^{\text{x}}$

Show that the vectors $2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$ and $-4\hat{\text{i}}+6\hat{\text{j}}-8\hat{\text{k}}$ are collinear.

An insurance company insured $3000$ scooters, $4000$ cars and $5000$ trucks. The probabilities of the accident involving a scooter, a car and a truck are $0.02, 0.03$ and $0.04$ respectively. One of the insured vehicles meet with an accident. Find the probability that it is a,

Scooter.
Car.
Truck.