Tag Archives: Integration
$\scriptstyle f(x) = {\left( {\frac{x}{\pi }} \right)^x} + {\left( {\frac{\pi }{x}} \right)^x}$
Integration of Infinite Series$\int {{{\left( {\frac{1}{{2!}} – \frac{1}{{3!}} + \frac{1}{{4!}} – \frac{1}{{5!}} + ………} \right)}^x}dx} = ?$
We have, ${e^{ – 1}} = 1 + ( – 1) + \frac{{{{( – 1)}^2}}}{{2!}} + \frac{{{{( – 1)}^3}}}{{3!}} + \frac{{{{( – 1)}^4}}}{{4!}} + ……….$
The given integral = $\int {{{({e^{ – 1}})}^x}dx} = \int {{e^{ – x}}dx} = – {e^{ – x}} + C$
$\int {\frac{{{{\sec }^2}x}}{{{{(\sec x + \tan x)}^5}}}dx} $
$\int\limits_0^{ – 2a} {f(x)dx} = ………$
If the system of equations 2x – y + z =0, x- 2y + z = 0 and ax – y + 2z = 0 has infinitely many solutions and f(x) is continuous function satisfying f(x)+f(x+5) = 2, then $\int\limits_0^{ – 2a} {f(x)dx}$ is equal to
a) 0
b) 5
c) a
d) –2a
Solution
