00:01
In this problem we are asked to evaluate the definite integral, integral 2 to 6, x over x squared plus 6 times x plus 13 d x.
00:14
So here let us make use of the substitution.
00:17
Consider the denominator to be equal to u.
00:20
So we have x squared plus 6 times x plus 13.
00:23
Differentiating this we get du to be equal to 2 times x plus 6 times d x.
00:31
So here let us first consider just the integrant which is x divided by x squared plus 6 times x plus 13.
00:40
Multiplying the numerator and the denominator with 2, we get 2 times x divided by 2 times x squared plus 6 times x plus 13.
00:51
Next we add and subtract 6.
00:54
So we get 2 times x plus 6 minus 6 in the numerator divided by 2 times x squared plus 6 plus 6 plus 6 times x squared plus 6 times x plus 13.
01:03
Now let us split this as 2x plus 6 divided by 2 times x squared plus 6 times x plus 13 and negative 6 over 2 times x squared plus 6x plus 13.
01:19
Now let us again consider the denominator's quadratic equation which is x squared plus 6x plus 13 and now we are going to complete the squares that is we can write x squared plus 6x plus 13 as x squared plus 2 times 3 times x plus 3 squared which is 9 minus 9 plus 13.
01:41
Simplifying this x squared plus 2 times 3 times x plus 9 can be written as x plus 3 the whole squared and 13 minus 9 is 4.
01:52
So this can be written as x plus 3 the whole squared plus 2 squared.
01:57
Now let us substitute this back in the second term.
02:00
We obtain the integrand to be equal to x divided by x squared plus 6 times x plus 13 equals to in the place of 2x plus 6 we can substitute d u also let us multiply d x with the integrant so we get d u over 2 times u minus 3 times d x divided by x plus 3 the whole squared plus 2 squared.
02:42
So now we see that the integrins can be evaluated using anti -derivatives.
02:50
So let us take integration on both the sides.
02:54
The integral of 1 over 2 u is 1 over 2 times natural log of u minus the integral of dx over x over x plus 3 the whole squared plus 2 squared plus 2 squared is 3 over 2 times tan inverse of x plus 3 divided by 2...