Evaluate: ^ _ 0 x x x + x dx - Vidyadip

Question

Evaluate:
$\int\limits^{\pi}_{0} \frac{\text{x} \tan \text{x}}{\sec \text{x} + \tan \text{x}}\text{dx}$

✓

Answer

$\text{I} = \int\limits^{\pi}_{0} \frac{\text{x} \tan \text{x}}{\sec \text{x} + \tan \text{x}} \text{dx} = \int\limits^{\pi}_{0} \frac{(\pi - \text{x)} \tan \text{x}}{\sec \text{x} + \tan \text{x}} \text{dx}$
$\Rightarrow \text{2I} = \pi \int\limits^{\pi}_{0} \frac{{\tan \text{x}}}{\sec {\text{x} + \tan \text{x}}} \text{dx} = \pi \int\limits^{\pi}_{0} \tan \text{x} (\sec \text{x} - \tan \text{x}) \text{dx}$
$\text{I} = \frac{\pi}{2} \int\limits^{\pi}_{0} {(\sec \text{x} \tan \text{x} - \sec^{2} \text{x + 1) dx}}$
$= \frac{\pi}{2} [ \sec \text{x} - \tan \text{x + x}]^{\pi}_{0}$
$= \frac{\pi(\pi - 2)}{2}$

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 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 man rides his motorcycle at the speed of 50km/ hour. He has to spend Rs. 2 per km on petrol. If he rides it at a faster speed of 80km/ hour, the petrol cost increases to Rs. 3 per km. He has atmost Rs. 120 to spend on petrol and one hour’s time. He wishes to find the maximum distance that he can travel.
Express this problem as a linear programming problem.

$\int\frac{\text{x + 2}}{\text{2x}^{2}+\text{6x + 5}}\text{dx}$.

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}1^2&2^2&3^2&4^2\\2^2&3^2&4^2&5^2\\3^2&4^2&5^2&6^2\\4^2&5^2&6^2&7^2\end{vmatrix}$

To confirm that, Is every real number in $R$. $R =\{(a, b): a, b \in R$ and $a-b+\sqrt{3} \in S \}$ where $S$ is set of all irrational numbers, defined, $R$ is reflexive, symmetric and transitive.

If y = $\frac{\sin^{-1}\text{x}}{\sqrt{\text{1 - x}^{2}}}$ show that
$(1-\text{x}^{2})\frac{\text{d}^{2}\text{y}}{\text{dx}}-\text{3x}\frac{\text{dy}}{\text{dx}}-\text{y}=0.$

If $\text{f}\text{(x)}=\begin{cases}\frac{1-\cos\text{kx}}{\text{x}\sin\text{x}}, & \text{x} \neq 0\\\frac{1}{2}, & \text{x}= 0\end{cases}$ is continuous at x = 0. find k.

$\text{Evaluate}\int\limits^{\pi}_{0} e^{2x} . \sin\big(\frac{\pi}{4} + x\big)\text{dx}$

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

Using matrices, solve the following system of equations:

$x + 2y - 3z = 6$

$3x + 2y - 2z =3$

$2x - y + z = 2$