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Evaluate the given integral.$$\int \frac{10 x^{4}-3 x^{2}+3 x+4}{4 x^{5}-2 x^{3}+3 x^{2}+8 x+5} d x$$

$$\frac{1}{2} \ln \left|3 x^{2}+4 x+1\right|+c$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

Oregon State University

Harvey Mudd College

Baylor University

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

04:44

Evaluate the integral.…

05:38

04:31

02:20

Evaluate the given integra…

the derivative of the polynomial and the denominator of this rational function is similar to the polynomial and the numerator of dysfunction. So this suggests, um that we apply a use substitution by taking you to be the function in the denominator. And hopefully when we take the derivative of this function, we can rewrite it in terms of the function in the numerator. That way we can replace d'you by the function that way that we can replace the function and the numerator times DX by do you So if you is equal to dysfunction, then d u d e X. That's equal to 20 times X to the fourth minus six X squared plus six acts plus eight and we can actually factor out eight to. So this becomes two times 10 to the 10 times X to the fourth minus three X squared plus three acts plus four, which is what we have for the function and the numerator of our integral. So from here we can write, do you over to is equal to the function in the numerator times dx and now we can rewrite this inter groom as one half times the integral of one over you, which is equal to one half times Ellen of the absolute value of you. And now we can substitute the function and the denominator for you. So this becomes Ellen of the absolute value of four x to the fifth minus to execute plus three x squared plus eight x plus five plus C, and that completes the problem.

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