00:01
In this question, we are asked to calculate the given integral.
00:04
And the first step would be to rewrite x to the 4 plus x squared plus 5 as a complete square.
00:10
We need to complete squares.
00:12
And to do that, we want to add a number and subtract a number such that the expression becomes x squared plus a squared plus some number.
00:28
Let's call it b.
00:29
This is our goal.
00:32
And first let's expand x squared plus a squared by the square of a sum formula that's going to be x to the fourth plus 2a x squared plus a squared note that the coefficient in front of x squared equals to 2a and in our formula the coefficient in front of x squared equals to 1 that means that 2a must be equal to 1 therefore a must be be equal to one half.
01:06
That means that a squared must be equal to one quarter, which means we need to add and subtract one quarter to the original formula.
01:23
And then we can rewrite that as x squared plus one half squared, and 5 minus one quarter equals to 19 quarters.
01:37
And that means that we can rewrite the original integral of 9x over x squared, sorry, x to the fourth.
01:47
Plus x squared plus 5 dx as the integral of 9x divided by x squared plus 1 half squared plus 19 quarters now we will make a trigonometric substitution i apologize we will make a u substitution first u equals to x squared plus 1 half then d u equals to 2x d x and after that substitution we will get the integral of 9 and x becomes the squared of u minus 1 half divided by u squared plus 19 quarters sorry i apologize we don't need u minus 1 half x d x becomes d u over 2 so we'll get 9 and multiplied by d u over 2 so this is the integral after the u substitution.
03:19
Let's factor out the constants...