# Integration with fraction and square root

• Dell
In summary, the conversation suggests using a substitution u = -4x^2 + 20x - 9 and breaking the integral into two parts. The first integral can be solved using a substitution and the second one may be solved using a trig substitution.
Dell
i am given this function and need to integrate it

$$\frac{4x+7}{\sqrt{-4x^2+20x-9}}$$

i have been trying to intergrate it by calling something T, preferably the expression under the sqrd root, or part thereof (-4x2+20x=T) but i can't find the way to do this, can't find the best expression that fits both the denominator and numerator of the fraction.. any other ideas??

Last edited:
Try u = -4x^2 + 20x - 9, and du = (-8x + 20)dx (I like u for substitutions better than T.)

Your numerator is 4x + 7 = 4x -10 + 17, so your integral can be rewritten as two integrals.

$$\int \frac{4x - 10}{\sqrt{-4x^2 + 20x - 9}}dx + \int \frac{17}{\sqrt{-4x^2 + 20x - 9}}dx$$

Applying the substitution in the first integral, we have $-1/2\int du/u^{1/2}$. The second one is probably amenable to a trig substitution.

## 1. How do you integrate a fraction?

To integrate a fraction, you first need to rewrite it in the form of a polynomial. Then, you can use the power rule or substitution method to solve the integral. For example, if you have the fraction 1/x, you can rewrite it as x^-1 and use the power rule to integrate.

## 2. Can square roots be integrated?

Yes, square roots can be integrated using the substitution method. This involves choosing a suitable substitution to rewrite the integral in a simpler form. For example, if you have the square root of x, you can substitute u = x^(1/2) to simplify the integral.

## 3. What is the general formula for integrating fractions and square roots?

The general formula for integrating fractions and square roots is not a one-size-fits-all solution. It depends on the specific fraction or square root being integrated. However, some commonly used integration techniques for these functions include the power rule, substitution method, and partial fractions.

## 4. How do you integrate a fraction with a variable in the denominator?

To integrate a fraction with a variable in the denominator, you can use the partial fractions method. This involves breaking down the fraction into simpler fractions and then integrating each part separately. For example, if you have the fraction 1/(x+1), you can rewrite it as 1/x + 1/(x+1) and integrate each part individually.

## 5. Can fractions and square roots be integrated together?

Yes, fractions and square roots can be integrated together. The key is to first simplify the expression as much as possible by using algebraic manipulations or substitution. Then, you can use the appropriate integration technique to solve the integral. It may also be helpful to break down the expression into smaller parts and integrate each part separately.

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