Evaluate the following integrals: ( ^ -1 x^2)x dx - Vidyadip

Question

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

✓

Answer

Let $\text{I}=\int(\tan^{-1}\text{x}^2)\text{x dx}$
Let $\text{x}^2=\text{t}$
$2\text{x dx = dt}$
$\text{I}=\frac{1}{2}\int\tan^{-1}\text{t dt}$
$=\frac{1}{2}\int1\tan^{-1}\text{t dt}$
$=\frac{1}{2}\Big[\tan^{-1}\text{t}\int\text{dt}-\Big(\int\frac{1}{1+\text{t}^2}\int\text{dt}\Big)\text{dt}\Big]$
$=\frac{1}{2}\Big[\text{t}\tan^{-1}\text{t}-\int\frac{\text{t}}{1+\text{t}^2}\text{dt}\Big]$
$=\frac{1}{2}\text{t}\tan^{-1}\text{t}-\frac{1}{4}\int\frac{2\text{t}}{1+\text{t}^2}\text{dt}$
$=\frac{1}{2}\text{t}\tan^{-1}\text{t}-\frac{1}{4}\log\big|1+\text{t}^2\big|+\text{C}$
$\text{I}=\frac{1}{2}\text{x}^2\tan^{-1}\text{x}^2-\frac{1}{4}\log\big|1+\text{x}^4\big|+\text{C}$

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 "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

A man 2 metres high walks at a uniform speed of 6km/h away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases.

If $\text{A}=\begin{bmatrix}3&1\\-1&2\end{bmatrix},$ show that $A^2 - 5A + 7I = 0$ use this to find $A^4.$

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

The perimeter of a triangle is 10 cm. If one of the sides is 4 cm. What are the other two sides of the triangle for its maximum area?

A wholesale dealer deals in two kinds, $A$ and $B$ (say) of mixture of nuts. Each kg of mixture $A$ contains 60 grams of almonds, 30 grams of cashew nuts and 30 grams of hazel nuts. Each kg of mixture B contains 30 grams of almonds, 60 grams of cashew nuts and 180 grams of hazel nuts. The remainder of both mixtures is per nuts. The dealer is contemplating to use mixtures A and B to make a bag which will contain at least 240 grams of almonds, 300 grams of cashew nuts and 540 grams of hazel nuts. Mixture $A$ costs Rs. 8 per kg. and mixture $B$ costs Rs. 12 per kg. Assuming that mixtures $A$ and $B$ are uniform, use graphical method to determine the number of kg . of each mixture which he should use to minimise the cost of the bag.

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

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

Integrate the following w. r. t. x:

$\frac{5 \cdot e^x}{\left(e^x+1\right)\left(e^{2 x}+9\right)}$

Divide $15$ into two parts such that the square of one multiplied with the cube of the other is minimum.

Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix}2 & 5 \\ 1 & 3 \end{bmatrix}$