Download App

Definite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
If f is an integrable function such that f(2a - x) = f(x), then prove that:
$\int\limits^{2\text{a}}_0\text{f(x)}\text{dx}=2\int\limits^{\text{a}}_0\text{f(x)}\text{dx}$
✓

Answer

Let $\text{I}=\int\limits^{2\text{a}}_0\text{f(x)}\text{dx}$
By Additive property
$\text{I}=\int\limits^{\text{a}}_0\text{f(x)}\text{dx}+\int\limits^{2\text{a}}_0\text{f(x)}\text{dx}$
Consider the integral $\int\limits^{2\text{a}}_0\text{f(x)}\text{dx}$
Let $\text{x}=2\text{a}-\text{t},$ then $\text{dx}=-\text{dt}$
When $\text{x}=\text{a},\text{ t}=\text{a},\text{ x}=2\text{x},\text{t}=0$
Hence $\int\limits^{2\text{a}}_0\text{f(x)}\text{dx}=-\int\limits^{0}_\text{a}\text{f(2a}-\text{t})\text{dt}$
$=\int\limits_{0}^\text{a}\text{f(2a}-\text{t})\text{dt}$
$=\int\limits_{0}^\text{a}\text{f(2a}-\text{x})\text{dx}$ (Changeing the varible)
Therefore,
$\text{I}=\int\limits_{0}^\text{a}\text{f(x})\text{dx}+\int\limits_{0}^\text{a}\text{f(2a}-\text{x})\text{dx}$
$=\int\limits_{0}^\text{a}\text{f(x})\text{dx}+\int\limits_{0}^\text{a}\text{f(x})\text{dx}$ $\Bigg[\text{Given}\int\limits_{0}^\text{a}\text{f(x})\text{dx}+\int\limits_{0}^\text{a}\text{f(2a}-\text{x})\text{dx}\Bigg]$
$=2\int\limits_{0}^\text{a}\text{f(x})\text{dx}$

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

Solve the following differential equations:$\cos\text{y}\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}},\text{y}(0)=\frac{\pi}{2}$
→
Show that $\text{f}\text{(x)}=\begin{cases}\frac{\sin 3\text{x}}{\tan2\text{x}},&\text{if } \text{x}<0\\\frac{3}{2},&\text{if }\text{x} = 0\\\frac{\log(1+3\text{x})}{\text{e}^{2\text{x}}},&\text{if}\text{ x}>0\end{cases}$ is discontinuous at x = 0.
→
Find the coordinates of the foot of perpendicular and perpendicular distance from the point $\text{ P(4, 3, 2)}$ to the plane $x + by+ 3z = 2.$ Also, find the image of P in the plane.
→
Solve the following system of equations by matrix method:
2x + y + z = 2
x + 3y − z = 5
3x + y − 2z = 6
→
An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit?
→
A letter is known to have come either from LONDON or CLIFTON. On the envelope just two consecutive letters ON are visible. What is the probability that the letter has come from,CLIFTON?
→
Find the angle between the pairs of lines with direction ratios proportional to
$2, 2, -2$ and $4, 1, 8$
→
$\overrightarrow{\text{n}}$ is a vector of magnitude $\sqrt{3}$ and is equally inclined to an acute angle with the coordinate axes. Find the vector and cartesian form of the equation of a plane which passes through (2, 1, -1) and is normal to $\overrightarrow{\text{n}}$
→
Find the corrdinates of the points P where the line throught A(3, -4,-5) and B(2, -3, 1) crosses the plane passing throught three points L(2, 2, 1), M(3, 0, 1) and N(4, -1, 0). Also, find the ratio in which P diveides the line segment AB.
→
The equation of tangent at (2, 3) on the curve $y^{2} = \text{ax}^{3} + \text{b is y = 4x - 5}. $ Find the value of a and b.
→