Question Video: Finding the Integration of a Rational Function by Completing the Square | Nagwa Question Video: Finding the Integration of a Rational Function by Completing the Square | Nagwa

# Question Video: Finding the Integration of a Rational Function by Completing the Square Mathematics • Higher Education

Evaluate โซ d๐ฅ/(๐ฅยฒ โ ๐ฅ + 1).

02:14

### Video Transcript

Evaluate the indefinite integral of one over ๐ฅ squared minus ๐ฅ plus one with respect to ๐ฅ.

Now, this isnโt a nice function to integrate at all. So weโre going to need to do something a little bit clever. Itโs certainly not the product of two functions. So weโre not going to use integration by parts. But if we do something special to the denominator, we can actually use integration by substitution. Weโre going to complete the square of the denominator of the expression ๐ฅ squared minus ๐ฅ plus one. Remember, we halve the coefficient of ๐ฅ. Here, thatโs negative one, so half of that is negative one-half. We, therefore, have ๐ฅ minus a half all squared in the brackets. Negative a half squared is one-quarter, so we subtract that one quarter. And we see that our expression is equivalent to ๐ฅ minus a half all squared plus three quarters. And now, this is the integral that weโre looking to evaluate.

Next, we need to spot that we know that indefinite integral of ๐ over ๐ squared plus ๐ฅ squared. Itโs the inverse tan of ๐ฅ over ๐. So to ensure that our function looks a little like this, weโre going to perform a substitution. Weโre going to let ๐ฅ minus a half be equal to ๐ข. Then this part will be ๐ข squared. The derivative of ๐ฅ minus one-half is one. So d๐ข by d๐ฅ equals one, which means that d๐ข is equal to d๐ฅ. So we can replace d๐ฅ with d๐ข and ๐ฅ minus a half with ๐ข. And we see that weโre actually looking to find the indefinite integral of one over ๐ข squared plus three-quarters.

Now, this still doesnโt quite look like what weโre after. We need it to be ๐ squared on the denominator. Well, three-quarters is the same as the square root of three quarters squared. So ๐ here is equal to the square root of three-quarters. And of course, since the numerates of our fraction is one and not the square root of three-quarters, the integral is one divided by the square root of three-quarters times the inverse tan of ๐ข over the square root of three quarters plus ๐. One divided by the square root of three quarters is two root three over three. And then we go back to our substitution ๐ข equals ๐ฅ minus a half. And we replace that in our result. And finally, we distribute our parentheses. The indefinite integral of one over ๐ฅ squared minus ๐ฅ plus one with respect to ๐ฅ is two root three over three times the inverse tan of root three over three times two ๐ฅ minus one plus the constant of integration ๐.

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