Find indefinite integral. ∫(3 x)/(√(2 x+3)) d x

Name: Problem 12, Chapter 7: Applied Calculus for the MLSS A Brief Approach
Uploaded: 2021-05-16T08:47:08-08:00
Duration: 4 min 14 s
Channel: Naresh Bagrecha
Description: Find indefinite integral. ∫(3 x)/(√(2 x+3)) d x

Key Concepts

Integration by Substitution (u-Substitution)

Integration by substitution is a technique used to simplify integrals by making a change of variable. It involves setting a new variable to a function within the integrand, which often transforms the integral into a simpler form. This method is particularly useful when the integrand is a composite function where one part of the function is the derivative of another part.

Integration of Functions Involving Radical Expressions

When dealing with integrals that involve radical expressions, such as square roots, it often becomes necessary to use algebraic manipulation or substitution to simplify the integrand. Recognizing how to handle and simplify these expressions is a crucial skill, as it allows one to transform a more complex integral into a form that is easier to evaluate using standard techniques.

Indefinite Integral

An indefinite integral represents a family of functions whose derivative is the integrand. Unlike definite integrals, indefinite integrals include an arbitrary constant because differentiation of a constant yields zero. This concept is fundamental in calculus for recovering a function from its derivative.

Transcript

00:01 In the question we have to find the value of given indefinite integral that is integral 3x divided by square root of 2x plus 3 dx so for finding this first i will assume that u is equal to 3x so d u will become three times of d x and assume that dv is equal to 1 divided by square root of 2x plus 3 d x so we will become 3x so we will become 3 we will be equal to after integration both sides on both side i will get value that is a square root of 2x plus 3 because 1 divided by square root of 2 x plus 3 will be that is 2 times of square root of 2 x plus 3 divided by 2 so we are 2 to 2 to cancel out so we will be equal to square root of 2x plus 3 now i can can write the given indefinite integral in this form that is integral 3x is u divided by 1 multiplied by 1 divided by square root of 2 x plus 3 d x will be dv so for finding this i am using a formula of indication by part which is written as integral u dv is equal to uv minus integral v d u so this is the formula of integration by part now substitute all data then i can write that is integral u dv will be equal to integral 3x divided by square root of 2x plus 3d d x will be equal to uv means that is uv is 3x and v will be square root of 2x and v will be of 2x plus 3 minus integral v du means integral v will be square root of 2x plus 3 and du will be 3 d x so after simplify this then i can say that integral 3x divided by square root of 2x plus 3 d x will be equal to 3x times of square root of 2x plus 3 minus 3 is come outside and remain inside is that is integral the square root of 2x plus 3 d x so integration will be that is after taking integration i will get that is minus 3 times of 2x plus 3 to the power 3 divided by 2 divided by 3 divided by 2 multiplied by 2 plus c so after again simplify this then i get the value of indefinite integral that is integral 3x divided by square root of 2x plus 3 dx will be equal to 3x multiplied by square root of 2 x plus 3 minus 2x plus 3 to the power 3 over 2 here all the terms are cancelled out.

03:58 So remain is minus 2 x plus 3 to the power 3 over 2 plus c...

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