Question

Find \[\int {\dfrac{{dx}}{{5 - 8x - {x^2}}}} \]

Answer 1

Hint: Here, you can see it is a question of calculus and you have to integrate the given function. For that we should know what integration is. In mathematics, integration is defined as the calculation of an integral. It is basically the summation of discrete data. In this question we will apply different formulae of integral calculus to get the required answer.

Step wise solution:
Given data: To find \[\int {\dfrac{{dx}}{{5 - 8x - {x^2}}}} \]
To convert \[5 - 8x - {x^2}\] into the form :
\[
{A^2} - {X^2}\\ \]
Here \[,\,\,5 - 8x - {x^2} \]

Adding 16 and subtracting 16, we get
\[
 = 5 - (8x + {x^2}) + 16 - 16\\
Now,\\
 = 5 - (8x + {x^2} + 16 - 16)\\
 = 5 - (8x + {x^2} + 16) - 16
\]
You can see \[{x^2} + 8x + 16\] is in the form of \[{a^2} + 2ab + {b^{2\,}}\,\,where\,\,a = x,b = 4\,\,and\,\,2ab = 8x\]

i.e.:\[ 5 - ({x^2} + 2 \cdot 4x + {4^2}) - 16\\
 = 21 - {(x + 4)^2} \]

21 can be written as \[{(\sqrt {21} )^2}\] right?
So,
\[ = 5 - 8x - {x^2} = {(\sqrt {21} )^2} - {(x + 4)^2}\]
Converted in the form \[{A^2} - {X^2}where\,\,A = (\sqrt {21} )\,\,and\,\,X = x + 4\]
To calculate \[\int {\dfrac{{dx}}{{5 - 8x - {x^2}}}} \]
Suppose,
\[I = \int {\dfrac{{dx}}{{5 - 8x - {x^2}}}} \]
I can also be written as, \[I = \int {\dfrac{{dx}}{{{{(\sqrt {21} )}^2} - {{(x + 4)}^2}}}} \]
According to the formula of integration, we knew that,
\[I = \int {\dfrac{{dx}}{{{A^2} - {X^2}}}} = \dfrac{1}{{2A}}\log \left| {\dfrac{{A + X}}{{A - X}}} \right| + C\]
Here, you can see \[I = \int {\dfrac{{dx}}{{{{(\sqrt {21} )}^2} - {{(x + 4)}^2}}}} \] is also in the form of \[\int {\dfrac{{dx}}{{{A^2} - {X^2}}}} \]
So, applying the formula of integration in I , we get
\[
I = \dfrac{1}{{2\sqrt {21} }}\log \left| {\dfrac{{\sqrt {21} + (x + 4)}}{{\sqrt {21} - (x + 4)}}} \right| + C\\
 \Rightarrow I = \dfrac{1}{{2\sqrt {21} }}\log \left| {\dfrac{{\sqrt {21} + x + 4}}{{\sqrt {21} - x - 4}}} \right| + C\\
 \Rightarrow \int {\dfrac{{dx}}{{5 - 8x - {x^2}}}} = \dfrac{1}{{2\sqrt {21} }}\log \left| {\dfrac{{\sqrt {21} + x + 4}}{{\sqrt {21} - x - 4}}} \right| + C\]
     where C is constant of integration.

Note: Students often get confused with the formula of integration. Be careful when you use the formulae. To avoid mistakes.