Definite Integration

If $a_n=\frac{3}{4}-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\cdots \cdots(-1)^{n-1}\left(\frac{3}{4}\right)^n$ and $b_n=1-a_n$, then find the minimum natural number $n_0$ such that $b_n>a_n \forall n>n_0$

\(\text { The value of } 5050 \frac{\int_0^1\left(1-x^{50}\right)^{100} d x}{\int_0^{\frac{1}{1}}\left(1-x^{50}\right)^{101} d x} \text { is : }\)

\(\int\limits_{ - 2}^0 {\left\{ {{x^3} + 3{x^2} + 3x + 3 + \left( {x + 1} \right)\cos \left( {x + 1} \right)} \right\}\,\,dx}\) is equal to

Evatuate:
\(\int_\limits{0}^{\pi} e^{|\cos x|}\left[2 \sin \left(\frac{1}{2} \cos x\right)+3 \cos \left(\frac{1}{2} \cos x\right)\right] \sin x ~d x\)

Evaluate \(\,\int\limits_0^\pi {{e^{\left| {\cos x} \right|}}} \left( {2\sin \left( {{1 \over 2}\cos x} \right) + 3\cos \left( {{1 \over 2}\cos x} \right)} \right)\sin x\,\,dx\)

If \(f(x)\) is differentiable and \(\int\limits_0^{{t^2}} {xf\left( x \right)dx = {2 \over 5}{t^5},}\) then \(f\left( {{4 \over {25}}} \right)\) equals

The value of the integral \(\int\limits_0^1 {\sqrt {{{1 - x} \over {1 + x}}} dx}\) is

If \(y\left( x \right) = \int\limits_{{x^2}/16}^{{x^2}} {{{\cos x\cos \sqrt \theta } \over {1 + {{\sin }^2}\sqrt \theta }}d\theta ,}\) then find \({{dy} \over {dx}}\) at \(x = \pi\)

Find the value of \(\int\limits_{ - \pi /3}^{\pi /3} {{{\pi + 4{x^3}} \over {2 - \cos \left( {\left| x \right| + {\pi \over 3}} \right)}}dx}\)

If \(l\left( {m,n} \right) = \int\limits_0^1 {{t^m}{{\left( {1 + t} \right)}^n}dt,}\) then the expression for \(l(m, n)\) in terms of \(l(m+n, n-1)\) is

If \(f\left( x \right) = \int\limits_{{x^2}}^{{x^2} + 1} {{e^{ - {t^2}}}} dt,\) then \(f(x)\) increases in

If \(f\) is an even function then prove that
\(\int\limits_0^{\pi /2} {f\left( {\cos 2x} \right)\cos x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\sin 2x} \right)\cos x\,dx.}\)

The integral \(\int\limits_{ - 1/2}^{1/2} {\left( {\left[ x \right] + \ell n\left( {{{1 + x} \over {1 - x}}} \right)} \right)dx}\) equal to

Let \(T>0\) be a fixed real number . Suppose \(f\) is a continuous
function such that for all \(x \in R\), \(f\left( {x + T} \right) = f\left( x \right)\).
If \(I = \int\limits_0^T {f\left( x \right)dx}\) then the value of $$\int\limits...

Let \(T>0\) be a fixed real number . Suppose \(f\) is a continuous
function such that for all \(x \in R\), \(f\left( {x + T} \right) = f\left( x \right)\).
If \(I = \int\limits_0^T {f\left( x \right)dx}\) then the value of $$\int\limits...

The value of \(\int\limits_{ - \pi }^\pi {{{{{\cos }^2}x} \over {1 + {a^x}}}dx,\,a > 0,}\) is

The value of the integral \(\int\limits_{{e^{ - 1}}}^{{e^2}} {\left| {{{{{\log }_e}x} \over x}} \right|dx}\) is :

Let \(g\left( x \right) = \int\limits_0^x {f\left( t \right)dt,}\) where f is such that
\({1 \over 2} \le f\left( t \right) \le 1,\) for \(t \in \left[ {0,1} \right]\) and \(\,0 \le f\left( t \right) \le {1 \over 2},\) for $$t \in \left[ ...

If \(f\left( x \right) = \left\{ {\matrix{ {{e^{\cos x}}\sin x,} & {for\,\,\left| x \right| \le 2} \cr {2,} & {otherwise,} \cr } } \right.\) then \(\int\limits_{ - 2}^3 {f\left( x \right)dx = }\)

For \(x>0,\) let \(f\left( x \right) = \int\limits_e^x {{{\ln t} \over {1 + t}}dt.}\) Find the function
\(f\left( x \right) + f\left( {{1 \over x}} \right)\) and show that $$f\left( e \right) + f\left( {{1 \over e}} \right) = {1 \over 2}....

\(\int\limits_{\pi /4}^{3\pi /4} {{{dx} \over {1 + \cos x}}}\) is equal to

Integrate \(\int\limits_0^\pi {{{{e^{\cos x}}} \over {{e^{\cos x}} + {e^{ - \cos x}}}}\,dx.}\)

If for a real number \(y\), \(\left[ y \right]\) is the greatest integer less than or
equal to \(y\), then the value of the integral \(\int\limits_{\pi /2}^{3\pi /2} {\left[ {2\sin x} \right]dx}\) is

Let \(f\left( x \right) = x - \left[ x \right],\) for every real number \(x\), where \(\left[ x \right]\) is the integral part of \(x\). Then \(\int_{ - 1}^1 {f\left( x \right)\,dx}\) is

Prove that \(\int_0^1 {{{\tan }^{ - 1}}} \,\left( {{1 \over {1 - x + {x^2}}}} \right)dx = 2\int_0^1 {{{\tan }^{ - 1}}} \,x\,dx.\)
Hence or otherwise, evaluate the integral
\(\int_0^1 {{{\tan }^{ - 1}}\left( {1 - x + {x^2}} \right)dx.}\)

If \(\int_0^x {f\left( t \right)dt = x + \int_x^1 {t\,\,f\left( t \right)\,\,dt,} }\) then the value of \(f(1)\) is

The value of \(\int_1^{{e^{37}}} {{{\pi \sin \left( {\pi In\,x} \right)} \over x}\,dx}\) is ...............

Determine the value of \(\int_\pi ^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} \,dx.\)

Let \({d \over {dx}}\,F\left( x \right) = {{{e^{\sin x}}} \over x},\,x > 0.\) If \(\int_1^4 {{{2{e^{\sin {x^2}}}} \over x}} \,\,dx = F\left( k \right) - F\left( 1 \right)\)
then one of the possible values of \(k\) is ............

If for nonzero \(x\), \(af(x)+\) \(bf\left( {{1 \over x}} \right) = {1 \over x} - 5\) where \(a \ne b,\) then
\(\int_1^2 {f\left( x \right)dx} = .......\)

For \(n>0,\) \(\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx =\)

If \(f\left( x \right)\,\,\, = \,\,\,A\sin \left( {{{\pi x} \over 2}} \right)\,\,\, + \,\,\,B,\,\,\,f'\left( {{1 \over 2}} \right) = \sqrt 2\) and
\(\int\limits_0^1 {f\left( x \right)dx = {{2A} \over \pi },}\) then constants \(A\) and $$...

The value of \(\int\limits_\pi ^{2\pi } {\left[ {2\,\sin x} \right]\,dx}\) where [ . ] represents the greatest integer function is

Evaluate the definite integral :
\($\int\limits_{ - 1/\sqrt 3 }^{1/\sqrt 3 } {\left( {{{{x^4}} \over {1 - {x^4}}}} \right){{\cos }^{ - 1}}\left( {{{2x} \over {1 + {x^2}}}} \right)} dx\)$

Let \({I_m} = \int\limits_0^\pi {{{1 - \cos mx} \over {1 - \cos x}}} dx.\) Use mathematical induction to prove that \({I_m} = m\,\pi ,m = 0,1,2,........\)

The value of \(\int\limits_2^3 {{{\sqrt x } \over {\sqrt {3 - x} + \sqrt x }}} dx\) is ...........

Show that \(\int\limits_0^{n\pi + v} {\left| {\sin x} \right|dx = 2n + 1 - \cos \,v}\) where \(n\) is a positive integer and \(\,0 \le v < \pi .\)

The value of \(\int\limits_{\pi /4}^{3\pi /4} {{\phi \over {1 + \sin \phi }}d\phi }\) is ..............

The value of \(\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}}\) is

Evaluate \(\int_2^3 {{{2{x^5} + {x^4} - 2{x^3} + 2{x^2} + 1} \over {\left( {{x^2} + 1} \right)\left( {{x^4} - 1} \right)}}} dx.\)

Determine a positive integer \(n \le 5,\) such that
\($\int\limits_0^1 {{e^x}{{\left( {x - 1} \right)}^n}dx = 16 - 6e}\)$

Evaluate \(\,\int\limits_0^\pi {{{x\,\sin \,2x\,\sin \left( {{\pi \over 2}\cos x} \right)} \over {2x - \pi }}dx}\)

Show that \(\int\limits_0^{\pi /2} {f\left( {\sin 2x} \right)\sin x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\cos 2x} \right)\cos x\,dx}\)

Prove that for any positive integer \(k\),
\({{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos \left( {2k - 1} \right)x} \right]\)
Hence prove that $$\int\limits_0^{\pi /2} {\sin 2kx\,\cot \,x\,dx = {\pi \over 2}} ...

Let \(f:R \to R\) and \(\,\,g:R \to R\) be continuous functions. Then the value of the integral
$$\int\limits_{ - \pi /2}^{\pi /2} {\left[ {f\left( x \right) + f\left( { - x} \right)} \right]\left[ {g\left( x \right) - g\left( { - x} \ri...

If \(f\) and \(g\) are continuous function on \(\left[ {0,a} \right]\) satisfying
\(f\left( x \right) = f\left( {a - x} \right)\) and \(g\left( x \right) + g\left( {a - x} \right) = 2,\)
then show that $$\int\limits_0^a {f\left( x \right)...

The value of \(\int\limits_{ - 2}^2 {\left| {1 - {x^2}} \right|dx}\) is ...............

Evaluate \(\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx}\)

The value of the integral \(\int\limits_0^{2a} {[{{f\left( x \right)} \over {\left\{ {f\left( x \right) + f\left( {2a - x} \right)} \right\}}}]\,dx}\) is equal to \(a\).

The integral \(\int\limits_0^{1.5} {\left[ {{x^2}} \right]dx,}\)
Where [ ] denotes the greatest integer function, equals .............