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Consider the integral $\int x^{n}\left(a x^{m}+b\right)^{r} d x,$ where, $a, b, m, n$ and $r$ are constants. What relationship should there be among these constants if the method of substitution will work to evaluate the integral?

If $n=m-1$ and $a m \neq 0$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 3

The Substitution Method

Integrals

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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.

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Okay in discussion. First of all three forms are given their tests from one. It is integration of dx divided by X square plus bx plus C. Okay. From two that is integration of mx plus and divided by a X square plus Bx plus C D X. Okay. And from three that is integration of dx divided by X squared plus bx plus C. Whole race to the power and we're and greater than zero. Okay. Three forms are there. And our question is we have to compare uh integration of dx divided by five minus X by four minus X squared by six. With one of them or either 12 or three. Okay. So we can see we will easily compared with from one that is integration of dx divided by he X squared plus bx plus C. Okay, so we will compare it well comparing the numerator the same both of side that is dx and the denominator coefficient of X esquire is a hair that is minus one by six. So it will be minus one by six. Okay? And be will be coefficient of X that is minus one by four. Okay. And sees constant value that is five here. Okay, So these are the values A, B and C and we have compared with from one. So it will be from one. Okay? And with value of constants that is equals two minus one by six, B equals two minus one by four and Sequels to five. And this will be the answer of our question. Thank you.

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