Evaluate the following integrals: x^6+1 x^2+1 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\text{x}^6+1}{\text{x}^2+1}\text{dx}$

✓

Answer

$\int\Big(\frac{\text{x}^6+1}{\text{x}^2+1}\Big)\text{dx}$
$=\int\bigg[\frac{(\text{x}^2)^3+1^3}{\text{x}^2+1}\bigg]\text{dx}$ $[\text{A}^3+\text{B}^3=(\text{A+B})(\text{A}^2-\text{AB}+\text{B}^2)]$
$=\int\frac{(\text{x}^2+1)(\text{x}^4-\text{x}^2+1)}{(\text{x}^2+1)}\text{dx}$
$=\int(\text{x}^4-\text{x}^2+1)\text{dx}$
$=\int\text{x}^4\text{dx}+\int\text{x}^2\text{dx}+\int1\text{dx}$
$=\frac{\text{x}^{4+1}}{4+1}-\frac{\text{x}^{2+1}}{2+1}+\text{x}+\text{C}$
$=\frac{\text{x}^5}{5}-\frac{\text{x}^3}{3}+\text{x}+\text{C}$

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 "3 Marks Question" from Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

A line passes throuth the point with position vector $2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$ and is in the direction of $3\hat{\text{i}}+4\hat{\text{j}}-5\hat{\text{k}}.$ Find equations of the line in vector and cartesian form.

Evaluate the following definite integrals:
$\int_{0}^\limits{\pi}\Big(\sin^2\frac{\text{x}}{2}-\cos^2\frac{\text{x}}{2}\Big)\text{dx}$

Consider f : N → N, g : N → N and h : N → R defined as f(x) = 2x, g(y) = 3y + 4 and $\text{h(z)}=\sin\text{z}$ for all $\text{x, y, z}\in\text{N.}$ Show that ho(gof) = (hog)of.

Prove that $\left[\begin{array}{lll}\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}} & \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}} & \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{a}}\end{array}\right]=2\left[\begin{array}{ll}\overrightarrow{\mathrm{a}} & \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\end{array}\right]$.

Write a value of $\int\frac{\text{a}^{\text{x}}}{3+\text{a}^{\text{x}}}\text{ dx}$

Write the derivative of $\sin\text{x}$ with respect to $\cos\text{x}$.

If $\cos ^{-1} \alpha+\cos ^{-1} \beta+\cos ^{-1} \gamma=3 \pi$ then find the value of $\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)$

Find the area of the parallelogram whose diagonals are:
$3\hat{\text{i}}+4\hat{\text{j}}$ and $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$

Eight coins are thrown simultaneously. Find the chance of obtaining at least six heads.

Evaluate the following integrals:
$\int\text{x}^3\sin\text{x}^4\text{dx}$