The value (round off to one decimal place) of \(\mathop \smallint \nolimits_{ - 1}^1 x\;{e^{\left| x \right|}}dx\) ______
Explanation
Given,
Function f(x) = x e|x|
Integral is -1 to 1.
If f(-x) = f(x) then the function is said to be even function
If f(-x) = - f(x) then the function is said to be odd function.
f(-x) = -x e|-x| = -x e|x| = - f(x)
∴ The given function is an odd function.
For an odd function:
\(\mathop \smallint \nolimits_{ - a}^a x\;{f(x)}dx\) = 0
For a even function
\(\mathop \smallint \nolimits_{ - a}^a x\;{f(x)}dx\) = 2 × \(\mathop \smallint \nolimits_{ 0}^a x\;{f(x)}dx\)
Now, as the function is odd
\(\mathop \smallint \nolimits_{ - 1}^1 x\;{e^{\left| x \right|}}dx\) = 0
Important Questions from Definite Integrals
The Legendre polynomials P n(x), n = 0, 1, 2, ..., satisfying the orthogonailty condition \(\int_{{\rm{ - 1}}}^{\rm{1}} {{{\rm{P}}_{\rm{n}}}\left( {\rm{x}} \right){{\rm{P}}_{\rm{m}}}} \left( {\rm{x}} \right){\rm{dx}}\,{\rm{ = }}\,\frac{{\rm{2}}}{{{\rm{2n + 1}}}}{{\rm{\delta }}_{{\rm{nm}}}}\) on the interval [-1, +1], may be defined by the Rodrigues formula P n(x) = \(\frac{{\rm{1}}}{{{{\rm{2}}^{\rm{n}}}{\rm{n!}}}}\frac{{{{\rm{d}}^{\rm{n}}}}}{{{\rm{d}}{{\rm{x}}^{\rm{n}}}}}{\left( {{{\rm{x}}^{\rm{2}}}{\rm{ - 1}}} \right)^{\rm{n}}}\) . The value of the definite integral \(\int_{{\rm{ - 1}}}^{\rm{1}} {\left( {{\rm{4 + 2x - 3}}{{\rm{x}}^{\rm{2}}}{\rm{ + 4}}{{\rm{x}}^{\rm{3}}}} \right){{\rm{P}}_{\rm{3}}}\left( {\rm{x}} \right){\rm{dx}}} \) is
A parametric curve is defined \(x = cos\left(\frac{\Pi t}{2}\right) , Y= sin\left(\frac{\Pi t}{2}\right)\) in the range of \(0\leq t\leq 1\) . It is rotated about X-axis by 360°.
‘What is the area of the surface generated?
if \(\displaystyle\int\dfrac{\sin x}{\sin (x-a)}dx=Ax+B\log |sin(x-a)|+ C\) where A, B and c are real constants then:
Which of the following is NOT a property of definite integral?
The value of integral \(x\over cos^2 x\) is equal to
