Evaluate the following integrals: _ 0 ^ + dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits_{0}^{\pi}\frac{\sin\text{x}}{\sin\text{x}+\cos\text{x}}\text{ dx}$

✓

Answer

$\int_{0}^\limits{\pi}\frac{\sin\text{x}}{\sin\text{x}+\cos\text{x}}\text{ dx}$
$=\frac{1}{2}\int_{0}^\limits{\pi}\frac{2\sin\text{x}}{\sin\text{x}+\cos\text{x}}\text{ dx}$
$=\frac{1}{2}\int_{0}^\limits{\pi}\frac{(\sin\text{x}+\cos\text{x})-(\cos\text{x}-\sin\text{x})}{\sin\text{x}+\cos\text{x}}\text{ dx}$
$=\frac{1}{2}\int_{0}^\limits{\pi}\text{dx}-\frac{1}{2}\int_{0}^\limits{\pi}\frac{\cos\text{x}-\sin\text{x}}{\sin\text{x}+\cos\text{x}}\text{ dx}$
$=\frac{1}{2}\big[\text{x}\big]^{\pi}_0-\frac{1}{2}\big[\log|\sin\text{x}+\cos\text{x}|\big]^{\pi}_0$
$=\frac{1}{2}\big[\pi-0\big]-\frac{1}{2}\big[\log1-\log1\big]$
$=\frac{\pi}{2}$

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 "5 Marks Questions" 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

For each of the differential equation in find the particular solution satisfying the given condition:$2\text{xy}+\text{y}^2-2\text{x}^2\frac{\text{dy}}{\text{dx}}=0;\ \text{y}=2\ \text{when x}=1$

Maximum Z = 3x + 4y Subject to$\text{x}+\text{y}\leq30000$
$\text{y}\leq12000$
$\text{x}\geq6000$
$\text{x}\geq\text{y}$
$\text{x},\text{y}\geq0$

Prove that the semi-vertical angle of the Right circular cone of given volume and least curved surface area is $\cot ^{-1}\sqrt{\text{2}}$

Evaluate the following integrals:
$\int\limits^4_{0}\big(|\text{x}|+|\text{x}+2|+|\text{x}+4|\big)\text{dx}$

Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹ 2,200. School Q wants to spend ₹ 3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each values is ₹ 1,200, using matrices, find the award money for each value.
Apart from these three values, suggest one more value which should be considered for award.

Solve the following for x and y.$\begin{bmatrix}3&-4\\9&2\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\end{bmatrix}=\begin{bmatrix}10\\2\end{bmatrix}$

find the area of the region in the first quadrant by the x-axis, the line $y = x$ and circle $x^2+ y^2 = 32$.

Show that the following systems of linear equations has infinite number of solutions and solve:

x - 2y + z = 0,

3x + 6y - 5z = 0

Find the equation of the tangent to the curve $\text{y}=\sqrt{3}\text{x}-2$ which is parallel to the 4x − 2y + 5 = 0.

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,
$\text{f}(\text{x})=\frac{\text{x}}{2}+\frac{2}{\text{x}}, \text{x}>0$