Evaluate the following integrals: ^ 2 _ 0 ^2x 1+3 ^3x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^\frac{\pi}{2}_{0}\frac{\cos^2\text{x}}{1+3\sin^3\text{x}}\text{ dx}$

✓

Answer

$\text{I}=\int^\limits\frac{\pi}{2}_{0}\frac{\cos^2\text{x}}{1+3\sin^3\text{x}}\text{ dx}$
$\text{I}=\int^\limits\frac{\pi}{2}_{0}\frac{\sec^2\text{x}}{\sec^2\text{x}\big(\sec^2\text{x}+3\tan^2\text{x}\big)}\text{ dx}$
Put $\tan\text{x}=\text{t}$
$\sec^2\text{x dx}=\text{dt}$
$\text{x}=0\Rightarrow\text{t}=0$ and $\text{x}=\frac{\pi}{2}\Rightarrow\text{t}=\infty$
$\Rightarrow\text{I}=\int^{\infty}\limits_0\frac{1}{\big(1+\text{t}^2\big)\big(1+4\text{t}^2\big)}\text{ dt}$
$\Rightarrow\text{I}=-\frac{1}{3}\int^{\infty}\limits_0\bigg[\frac{1}{(1+\text{t}^2)-{(1+4\text{t}^2)}}\bigg]\text{dt}$
$\Rightarrow\text{I}=-\frac{1}{3}\Big[\tan^{-1}\text{t}-2\tan^{-1}2\text{t}\Big]^{\infty}_0$
$\Rightarrow\text{I}=\frac{\pi}{6}$

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 "5 Marks Questions" from Definite Integrals→Definite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the value of $\lambda$ for which the lines $\frac{\text{x}-1}{1}=\frac{\text{y}-2}{2}=\frac{\text{z}+3}{\lambda^2}$ and $\frac{\text{x}-3}{1}=\frac{\text{y}-2}{\lambda^2}=\frac{\text{z}-1}{2}$ are coplanar.

If $\text{A}=\begin{bmatrix}3&-5\\-4&2\end{bmatrix},$ then find $A^2 - 5A - 14I$. Hence, obtain $A^3$.

A manufacturer has three machines installed in his factory. machines $I$ and $II$ are capable of being operated for at most $12$ hours whereas Machine $III$ must operate at least for $5$ hours a day. He produces only two items, each requiring the use of three machines. The number of hours required for producing one unit each of the items on the three machines is given in the following table:

Item	Number of hours required by the machine
	$I$	$II$	$III$
$A$	$1$	$2$	$1$
$B$	$2$	$1$	$\frac{5}{4}$

He makes a profit of $Rs. 6.00$ on item $A$ and $Rs. 4.00$ on item $B.$ Assuming that he can sell all that he produces, how many of each item should he produces so as to maximize his profit? Determine his maximum profit. Formulate this $\text{LPP}$ mathematically and then solve it.

Using properties of determinants, prove the following:$\begin{vmatrix} \text{x} &\text{x}^{2} & \text{1 + px}^{3} \\ \text{y} & \text{y}^{2} & \text{1 + py}^{3} \\ \text{z} & \text{z}^{2} & \text{1 + pz}^{3} \end{vmatrix}=\text{(1 + pxyz) (x - y)(y - z)(z - x)}$.

Prove that the curves $xy = 4$ and $x^2 + y^2 = 8$ touch each other.

Verify Rolle's theorem for the following function on the indicated intervals $f(x) = x^2 -4x + 3$ on $[1, 3]$

Find the approximate change in the surface area of a cube of side $x$ metres caused by decreasing the side by $1\%.$

If $\text{x}=3\sin\text{t}-\sin3\text{t},$ $\text{y}=3\cos-\cos3\text{t}$ find $\frac{\text{dy}}{\text{dx}}$ at $\text{t}=\frac{\pi}{3}.$

Solve the following determinant equations: $\begin{vmatrix}1&\text{x}&\text{x}^2\\1&\text{a}&\text{a}^2\\1&\text{b}&\text{b}^2\end{vmatrix}=0,\text{a}\neq\text{b}$

Prove that $\begin{vmatrix}a^2&bc&ac+c^2\\a^2+ab&b^2&ac\\ab&b^2+bc&c^2\end{vmatrix}=4a^2b^2c^2$