Question
Evaluate :
$
\int_0^{\frac{\pi}{4}}\left(\frac{\sin x+\cos x}{3+\sin 2 x}\right) d x
$
$
\int_0^{\frac{\pi}{4}}\left(\frac{\sin x+\cos x}{3+\sin 2 x}\right) d x
$
✓
Answer
Let
$
I=\int_0^{\frac{\pi}{4}} \frac{\sin x+\cos x}{3+\sin 2 x} d x
$
Here putting $\sin x-\cos x= t$
$
(\cos x+\sin x) d x=d t
$
So if $x=\frac{\pi}{4}$ then $t=0$ and if $x=0$ then $t=-1$
$
\begin{array}{l}
\begin{array}{l}
(\sin x-\cos x)^2=1-2 \sin x \cos x=1-\sin 2 x \\
\therefore \sin 2 x=1-(\sin x-\cos x)^2 \\
=1-t^2
\end{array} \\
\begin{aligned}
I & =\int_{-1}^0 \frac{d x}{3+\left(1-t^2\right)}=\int_{-1}^0 \frac{d t}{4-t^2}
\end{aligned} \\
I=\int_{-1}^0 \frac{d t}{(2)^2-t^2}=\frac{1}{2 \times 2}\left(\log \frac{2+t}{2-t}\right)_{-1}^0 \\
\quad\left(\because \int \frac{1}{a^2-x^2} d x=\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+c\right)
\end{array}
$
$
\begin{array}{l}
I=\frac{1}{4}\left(\log 1-\log \frac{1}{3}\right) \\
I=\frac{1}{4}\left(0-\log \frac{1}{3}\right)=\frac{1}{4} \log 3
\end{array}
$
$
I=\int_0^{\frac{\pi}{4}} \frac{\sin x+\cos x}{3+\sin 2 x} d x
$
Here putting $\sin x-\cos x= t$
$
(\cos x+\sin x) d x=d t
$
So if $x=\frac{\pi}{4}$ then $t=0$ and if $x=0$ then $t=-1$
$
\begin{array}{l}
\begin{array}{l}
(\sin x-\cos x)^2=1-2 \sin x \cos x=1-\sin 2 x \\
\therefore \sin 2 x=1-(\sin x-\cos x)^2 \\
=1-t^2
\end{array} \\
\begin{aligned}
I & =\int_{-1}^0 \frac{d x}{3+\left(1-t^2\right)}=\int_{-1}^0 \frac{d t}{4-t^2}
\end{aligned} \\
I=\int_{-1}^0 \frac{d t}{(2)^2-t^2}=\frac{1}{2 \times 2}\left(\log \frac{2+t}{2-t}\right)_{-1}^0 \\
\quad\left(\because \int \frac{1}{a^2-x^2} d x=\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+c\right)
\end{array}
$
$
\begin{array}{l}
I=\frac{1}{4}\left(\log 1-\log \frac{1}{3}\right) \\
I=\frac{1}{4}\left(0-\log \frac{1}{3}\right)=\frac{1}{4} \log 3
\end{array}
$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
A bag contains 3 red and 2 black balls. One ball is drawn from it at random. Its colour is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing,
- Two red balls,
- Two black balls,
- First red and second black ball.
Differentiate the following with respect to x: $\tan^{-1} \bigg(\frac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 + x + \sqrt{ 1 - x}}}\bigg)$
→Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner 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?
Tow godowns, A and B, have grain storage capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F, whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:
How should the supplies be transported in order that the transportation cost is minimum?
Evaluate the following integrals:
$\int\frac{1}{\text{x}^2-10\text{x}+34}\text{dx}$
→$\int\frac{1}{\text{x}^2-10\text{x}+34}\text{dx}$
Show that the semi-vertical angle of the cone of the maximum value and of given slant height is ${\tan ^{ - 1}}\sqrt 2$
→If y $ = \log\bigg[\text{x+}\sqrt{\text{x}^{2} + \text{a}^{2}}\bigg],\text {show that } (\text{x}^{2} + \text{a}^{2})\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} + \text{x}\frac{\text{dy}}{\text{dx}} = 0.$
→A man owns a field of area 1000 sq.m. He wants to plant fruit trees in it. He has a sum of Rs. 1400 to purchase young trees. He has the choice of two types of trees. Type A requires 10 sq.m of ground per tree and costs Rs. 20 per tree and type B requires 20 sq.m of ground per tree and costs Rs. 25 per tree. When fully grown, type A produces an average of 20kg of fruit which can be sold at a profit of Rs. 2.00 per kg and type B produces an average of 40kg of fruit which can be sold at a profit of Rs. 1.50 per kg. How many of each type should be planted to achieve maximum profit when the trees are fully grown? What is the maximum profit?
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\big(2\log\cos\text{x}-\log\sin2\text{x}\big)\text{dx}$
→$\int\limits^{\frac{\pi}{2}}_0\big(2\log\cos\text{x}-\log\sin2\text{x}\big)\text{dx}$