Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS4 Marks
Question
Evaluate the following integrals:
$\int\limits^{\pi}_0\frac{\text{x}\sin\text{x}}{1+\sin\text{x}}\text{ dx}$
✓

Answer

Let $\text{I}=\int\limits^{\pi}_0\frac{\text{x}\sin\text{x}}{1+\sin\text{x}}\text{ dx}\ ...(\text{i})$
$=\int\limits^{\pi}_0\frac{(\pi-\text{x})\sin(\pi-\text{x})}{1+\sin(\pi-\text{x})}\text{ dx}$
$=\int\limits^{\pi}_0\frac{(\pi-\text{x})\sin\text{x}}{1+\sin\text{x}}\text{ dx}\ ....(\text{ii})$
Adding (i) and (ii) we get
$2\text{I}=\int\limits^{\pi}_0(\text{x}+\pi-\text{x})\frac{\sin\text{x}}{1+\sin\text{x}}\text{ dx}$
$=\int\limits^{\pi}_0\frac{\pi\sin\text{x}}{1+\sin\text{x}}\text{ dx}$
$=\pi\int\limits^{\pi}_0\frac{1+\sin\text{x}-1}{1+\sin\text{x}}\text{ dx}$
$=\pi\int\limits^{\pi}_0\text{dx}-\pi\int\limits^{\pi}_0\frac{1}{1+\sin\text{x}}\text{ dx}$
$=\pi\int\limits^{\pi}_0\text{dx}-\pi\int\limits^{\pi}_0\frac{(1-\sin\text{x})}{(1+\sin\text{x})(1-\sin\text{x})}\text{ dx}$
$=\pi\int\limits^{\pi}_0\text{dx}-\pi\int\limits^{\pi}_0\frac{(1-\sin\text{x})}{1-\sin}$
$=\pi\int\limits^{\pi}_0\text{dx}-\pi\int\limits^{\pi}_0\big(\sec^2\text{x}-\sec\text{x}\tan\text{x}\big)\text{dx}$
$=\pi\big[\text{x}\big]^{\pi}_0-\pi\big[\tan\text{x}-\sec\text{x}\big]^{\pi}_0$
$=\pi^2-\pi(0+1-0+1)$
$=\pi^2-2\pi$
Hence, $\text{I}=\pi\Big(\frac{\pi}{2}-1\Big)$

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

Find the image of the point (1, 3, 4) in the plane 2x - y + z + 3 = 0.
→
Let $\text{A}=\begin{bmatrix}1&-1&0\\2&1&3\\1&2&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&2&3\\2&1&3\\0&1&1\end{bmatrix},$ Find AT, BT and verify that.

$(\text{A}\text{B})^\text{T}=\text{B}^\text{T}+\text{A}^\text{T}$

→
A box manufacturer makes large and small boxes from a large piece of cardboard. The large boxes require 4 sq. metre per box while the small boxes require 3 sq. metre per box. The manufacturer is required to make at least three large boxes and at least twice as many small boxes as large boxes. If 60 sq. metre of cardboard is in stock, and if the profits on the large and small boxes are Rs. 3 and Rs. 2 per box, how many of each should be made in order to maximize the total profit?
→
Prove that: $\big(\vec{\text{a}}-\vec{\text{b}}\big).\big\{\big(\vec{\text{b}}-\vec{\text{c}}\big)\big\}=0$
→
Two factories decided to award their employees for three values of (a) adaptable to new techniques, (b) careful and alert in difficult situations and (c) keeping clam in tense situations, at the rate of ₹ x, ₹ y and ₹ z per person respectively. The first factory decided to honuor respectively 2, 4 and 3 employees with a total prize money of ₹ 29000. The second factory decided to honuor respectively 5, 2 and 3 employees with the prize money of ₹ 30500. If the three prizes per person together cost ₹ 9500, then
  1.  Represent the above situation by matrix equation and form linear equation using matrix multiplication.
  2. Solve this equation by matrix method.
  3. Which values are reflected in the questions?
→
Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}|\text{x}-3|,&\text{if }\text{ x}\geq1\\\frac{\text{x}^2}{4}-\frac{3\text{x}}{2}+\frac{13}{4},&\text{if }\text{ x}<1\end{cases}$
→
Differentiate the following functions with respect to x:
$\sin^{-1}\Big\{\frac{\sqrt{1+\text{x}}+\sqrt{1-\text{x}}}{2}\Big\},0<\text{x}<1$
→
Find the matrix X for which:
$\begin{bmatrix}3 & 2 \\ 7 & 5 \end{bmatrix}\text{X}\begin{bmatrix} -1 & 1 \\ -2 & 1 \end{bmatrix}=\begin{bmatrix} 2 & -1 \\ 0 & 4 \end{bmatrix}$
→
Evaluate the following definite integrals:
$\int_{0}^\limits{1}\Big(\text{xe}^{\text{x}}+\cos\frac{\pi\text{x}}{4}\Big)\text{dx}$
→
Find the distance of the point P(3, 4, 4) from the point, where the line joining the points A(3, -4, -5) and B(2, -3, 1) intersects the plane 2x + y + z = 7.
→