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

Similar questions

A rectangular sheet of tin $45$cm by $24$cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?

Solve the following equation
$\text{x}\cos^2\text{y dx}=\text{y}\cos^2\text{x dx}$

Find the distance between the parallel planes $2x - y + 3z − 4 = 0$ and $6x - 3y + 9z + 13 = 0$.

Find the values of a and b, if the function f defined by
$\text{f}(x) = \begin{cases} x^{2} + 3x + \text{a} \text{ }, & x \leq 1\\ bx + 2 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ },& x > 1\\ \end{cases}$
is differentiable at x = 1.

Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of kings.

Let L be the set of all lines in xy plane and R be the relation in L. define as $s R =\left\{\left( L _1, L_2\right): L _1 \| L _2\right\}$ Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4

Find the value of 'a' for which the function f defined by
$\text{f}\text{(x)}=\begin{cases}\text{a}\sin\frac{\pi}{2}(\text{x}+1),& \text{x}\leq0 \\\frac{\tan\text{x-sin}\text{x}}{\text{x}^3} &\text{x} > 0\end{cases}$ is discontinuous at x = 0.

A shopkeeper sells three types of flower seeds $A_1, A_2$ and $A_3$. They are sold as a mixture where the proportions are 4 $: 4: 2$ respectively. The germination rates of the three types of seeds are $45 \%, 60 \%$ and $35 \%$. Calculate the probability:
i. Of a randomly chosen seed to germinate.
ii. That it will not germinate given that the seed is of type $A_3$.
iii. That it is of the type $A_2$ given that a randomly chosen seed does not germinate.

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
$\text{f}(\text{x})=\text{x}\sqrt{32-\text{x}^{2}}, -5\leq\text{x}\leq5$

Evaluate the following integrals:
$\int_{0}^\limits{1}\frac{1}{1+2\text{x}+2\text{x}^2+2\text{x}^3+\text{x}^4}\text{ dx}$