Question

Integrate x^9/(4x^2+1)^6

Answer 1

Answer:

I=∫x9(x2+4)6dx

Let x=2tant ; dx=2sec2tdt

I=∫210tan9tsec2tdt(4sec2t)6

=∫sin9tcostdt4

=sin10t40+C

But sint=xx2+4−−−−−√

I=x1040(x2+4)5+C

Step-by-step explanation:

Answer 2

Given:

Function f

$\bf \: \frac{ {x}^{9} }{(4 {x}^{2} + 1) ^{6} }$

To find :

Integration of above function f.

Solution:

The given function is :

$\bf \frac{ {x}^{9} }{(4 {x}^{2} + 1) ^{6} }$

Integrating this function :

$\bf \int \frac{ {x}^{9} }{(4 {x}^{2} + 1) ^{6} }$

Taking x out of the denominator

$\bf \int \frac{ {x}^{9} }{ {x}^{12} (4 + \frac{1}{ ({x}^{2}) }) ^{6} }$

cancelling power of x with numerator :

$\bf \int \frac{ 1 }{ {x}^{3} (4 + \frac{1}{ ({x}^{2}) }) ^{6} }$

Put

$(4 + \frac{1}{ {x}^{2} }) = t \\ \\ so \\ \frac{ - 2}{ ({x}^{3}) } = dt$

Then integration becomes :

$\bf \int \: \frac{dt}{2 {t}^{6} }$

Now integrating above equation, we get

$\bf \frac{1}{10 \times {t}^{5} } + c$

Putting the value of t from above

The integration of function f =

$\bf \: \frac{1}{10} (4 + \frac{1}{ {x}^{2} } ) ^{ - 5} + c$