Evaluate the following integrals: ^3 ^ -1 x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{x}^3\tan^{-1}\text{x dx}$

✓

Answer

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

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 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

If $\text{e}^{\text{x}}+\text{e}^{\text{y}}=\text{e}^{\text{x}+\text{y}},$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{e}^{\text{x}}(\text{e}^\text{y}-1)}{\text{e}^{\text{y}}(\text{e}^{\text{x}}-1)}$ or $\frac{\text{dy}}{\text{dx}}+\text{e}^{\text{y}-\text{x}}=0$

Find which of the function in Exercises 2 to 10 is continuous or discontinuos at the indicated point:
$\text{f(x)}=\begin{cases}\frac{\text{e}^{\frac{1}{\text{x}}}}{1+\text{e}^{\frac{1}{\text{x}}}},&\text{if x}\neq0\\0,&\text{if x }=\text{a}\end{cases}$
at x = 0

The cost of 4kg onion, 3kg wheat and 2kg rice is Rs. 60. The cost of 2kg onion, 4kg wheat and 6kg rice is Rs. 90. The cost of 6kg onion 2kg wheat and 3kg rice is Rs. 70. Find the cost of each item per kg by matrix method.

Differentiate the following functions with respect to x:
$\cos^{-1}\Big(\frac{1-\text{x}^{2\text{n}}}{1+\text{x}^{2\text{n}}}\Big), <\text{x}<\infty$

$\text{if } \vec{\text{a}} = 2\hat{\text{i}} + \hat{\text{j}} - \hat{\text{k}}, \vec{\text{b}} = 4\hat{\text{i}} - 7\hat{\text{j}} + \hat{\text{k}}, \text{find a vector } \vec{\text{c}} \text{ such that } \vec{\text{a}} \times \vec{\text{c}} \text{ and } \vec{\text{a }} . \vec{\text{c}} = 6.$

$\tan^{-1}\frac{2\text{a}\text{b}}{\text{a}^2-\text{b}^2}+\tan^{-1}\frac{2\text{xy}}{\text{x}^2-\text{y}^2}=\tan^{-1}\frac{2\alpha\beta}{\alpha^2-\beta^2},$

where $\alpha=\text{ax}-\text{by}$ and $\beta=\text{ay}+\text{bx}.$

If $\vec{\text{a}}=2\hat{\text{i}}+5\hat{\text{j}}-7\hat{\text{k}},\vec{\text{b}}=-3\hat{\text{i}}+4\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}},$ compute $\big(\vec{\text{a}}\times\vec{\text{b}}\big)\times\vec{\text{c}}$ and $\vec{\text{a}}\times\big(\vec{\text{b}}\times\vec{\text{c}}\big)$ and verify that these are not equal.

Find the angle between the lines whose direction cosines are given by the equations:
2l + 2m - n = 0, mn + ln + lm = 0

Is $|\sin\text{x}|$ differentible? What about $\cos|\text{x}|?$

Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector $2\hat{i} + 3\hat{j} + 4\hat{k}$ to the plane $\overrightarrow{\text{r}}. (2\hat{i} + \hat{j} + 3\hat{k}) - 26 = 0.$ Also find image of P in the plane.