Evaluate the following integrals: _ 0 ^ 1 x ^ -1 x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits_{0}^{1}\text{x}\tan^{-1}\text{x}\text{ dx} $

✓

Answer

We have,
$\int_{0}^\limits{1}\text{x}\tan^{-1}\text{x}\text{ dx}=\tan^{-1}\text{x}\int_{0}^\limits{1}\text{x dx}-\int_{0}^\limits{1}\big(\int\text{dx}\big)\frac{\text{d}}{\text{dx}}\big(\tan^{-1}\text{x}\big)\text{dx}$
$=\Big[\frac{\text{x}^2}{2}\tan^{-1}\text{x}\Big]^1_0-\frac{1}{2}\int_{0}^\limits{1}\frac{\text{x}^2}{1+\text{x}^2}\text{ dx}$
$=\Big[\frac{\text{x}^2}{2}\tan^{-1}\text{x}\Big]^1_0-\frac{1}{2}\int_{0}^\limits{1}\frac{1+\text{x}^2-1}{1+\text{x}^2}\text{ dx}$
$=\frac{1}{2}\Big(\frac{\pi}{4}\Big)-\frac{1}{2}\Bigg[\int_{0}^\limits{1}\text{dx}-\int_{0}^\limits{1}\frac{\text{dx}}{1+\text{x}^2}\Bigg]$
$=\frac{\pi}{8}-\frac{1}{2}\big[\text{x}-\tan^{-1}\text{x}\big]^{1}_0$
$=\frac{\pi}{8}-\frac{1}{2}\Big[1-\frac{\pi}{4}\Big]$
$=\frac{\pi}{8}-\frac{1}{2}+\frac{\pi}{8}$
$=\frac{\pi}{4}-\frac{1}{2}$

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 Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the general solution of the differential equation
y dx – (x + 2y²) dy = 0.

Evaluate $\int^{1}_{0}\text{e}^{2-3\text{x}}\text{dx}$ as a limit of a sum:

If $\text{x}\sin(\text{a}+\text{y})+\sin\text{a}\cos(\text{a}+\text{y})=0,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sin^2(\text{a}+\text{y})}{\sin\text{a}}.$

Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}+2\text{y}\tan\text{x}=\sin\text{x},\text{ y}=0,\text{ when x}=\frac{\pi}{3}$

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is $6\sqrt{3}\text{ r}$.

There are two types of fertilisers 'A' and 'B'. 'A' consists of 12 % nitrogen and 5 % phosphoric acid whereas 'B' consists of 4 % nitrogen and 5 % phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs 10 per kg and 'B' cost 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost.

Refer to Question 8. If the grower wants to maximize the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?

Solve the following equation for x:
$\tan^{-1}\frac{1}{4}+2\tan^{-1}\frac{1}{5}+\tan^{-1}\frac{1}{6}+\tan^{-1}\frac{1}{\text{x}}=\frac{\pi}{4}$

Solve the following differential equations:
$\text{y}\sqrt{1+\text{x}^2}+\text{x}\sqrt{1+\text{y}^2}\ \frac{\text{dy}}{\text{dx}}=0$

Prove that $\text{f}(\text{x})=\sin\text{x}+\sqrt{3}\cos\text{x}$ has maximum value at $\text{x}=\frac{\pi}{6}$.