Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS5 Marks
Question
If $\text{f}(\text{a}+\text{b}-\text{x})=\text{f(x)},$ then prove that, $\int\limits^{\text{b}}_{\text{a}}\text{xf}(\text{x})\text{dx}=\Big(\frac{\text{a}+\text{b}}{2}\Big)\int\limits^{\text{b}}_{\text{a}}\text{f}(\text{x})\text{dx}$
✓

Answer

$\int\limits^{\text{b}}_{\text{a}}\text{xf}(\text{x})\text{dx}=\int\limits^{\text{b}}_{\text{a}}\text{f}(\text{a}+\text{b}-\text{x})(\text{a}+\text{b}-\text{x}){}\text{dx}$ $\Bigg[\int\limits^{\text{b}}_\text{a}\text{f(x)}\text{dx}=\int\limits^{\text{b}}_\text{a}\text{f}(\text{a}+\text{b}-\text{x})\text{dx}\Bigg]$
$\Rightarrow\int\limits^{\text{b}}_{\text{a}}\text{xf}(\text{x})\text{dx}=\int\limits^{\text{b}}_{\text{a}}(\text{a}+\text{b}-\text{x})\text{f}(\text{x})\text{dx}$ $\big[\text{f}(\text{a}+\text{b}-\text{x})=\text{f}(\text{x})\big]$
$\Rightarrow\int\limits^{\text{b}}_{\text{a}}\text{xf}(\text{x})\text{dx}=\int\limits^{\text{b}}_{\text{a}}(\text{a}+\text{b})\text{f}(\text{x})\text{dx}-\int\limits^{\text{b}}_{\text{a}}\text{x}\text{f}(\text{x})\text{dx}$
$\Rightarrow2\int\limits^{\text{b}}_{\text{a}}\text{xf}(\text{x})\text{dx}=(\text{a}+\text{b})\int\limits^{\text{b}}_{\text{a}}\text{f}(\text{x})\text{dx}$
$\Rightarrow\int\limits^{\text{b}}_{\text{a}}\text{xf}(\text{x})\text{dx}=\Big(\frac{\text{a}+\text{b}}{2}\Big)\int\limits^{\text{b}}_{\text{a}}\text{f}(\text{x})\text{dx}$

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 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

If $\text{P}=\begin{bmatrix}\text{x}&0&0\\0&\text{y}&0\\0&0&\text{z}\end{bmatrix}$ and $\text{Q}=\begin{bmatrix}\text{a}&0&0\\0&\text{b}&0\\0&0&\text{c}\end{bmatrix},$ prove that $\text{PQ}=\begin{bmatrix}\text{xa}&0&0\\0&\text{y}\text{b}&0\\0&0&\text{zc}\end{bmatrix}=\text{QP}$
→
Evaluate the following integrals:$\int\frac{\text{x}+1}{\text{x}^2+\text{x}+3}\text{ dx}$
→
Find the area enclosed by the curve $\text{x}=3\cos\text{t}, \text{y}=2\sin\text{t}.$
→
Let f be a real function given by $\text{f(x)}=\sqrt{\text{x}-2}.$ Find the following:
(fofof)(38)
Also, show that fof ≠ $f^2$.
→
Show that the differential equation $x^{2} d y+\left(x y+y^{2}\right) d x=0$ is homogenous and find the particular solution, given that $y = 1$ when $x = 1$.
→
Evaluate the following:
$\begin{vmatrix}\text{a}+\text{x}&\text{y}&\text{z}\\\text{x}&\text{a}+\text{y}&\text{z}\\\text{x}&\text{y}&\text{a}+\text{z}\end{vmatrix}$
→
Find the equation of the plane passing through the line of intersection of the planes $\vec{\text{r}}.\Big(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\Big)=1\ \text{and}\ \vec{\text{r}}.\Big(2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}\Big)+4=0$ and parallel to x-axis.
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}-\text{y}=\text{xe}^{\text{x}}$
→
Show that the following systems of linear equations has infinite number of solutions and solve:
$2x + y - 2z = 0,$
$x - 2y + z = -2,$
$5x - 5y + z = -2$
→
Solve the following system of equations by matrix method:
$3x + 4y + 2z = 8$
$2y - 3z = 3$
$x - 2y + 6z = -2$
→