It looks good. Take the derivative to check it.KMcFadden said:Homework Statement
∫▒〖(4x^2+2√x+1)/(2x√x) dx〗
Homework Equations
The Attempt at a Solution
∫▒〖(4x^2+2√x+1)/(2x√x) dx〗
∫▒((4x^2)/(2x^(3/2) )+(2√x)/(2x^(3/2) )+1/(2x^(3/2) ))dx
2∫▒x^(1/2) dx+∫▒〖x^(-1) dx〗+1/2 ∫▒x^(-3/2) dx
2×2/3 x^(3/2)+lnx+1/2×(-2) x^(-1/2)+C
4/3 x^(3/2)+lnx-x^(-1/2)+C
4/3 √(x^3 )+lnx-1/√x+C
An indefinite integral is a mathematical concept that represents the antiderivative of a given function. It is a fundamental operation in calculus and is used to find the original function from its derivative.
To find the indefinite integral of a function, you must use integration techniques such as u-substitution, integration by parts, or trigonometric substitution. It is important to also include the constant of integration, as the indefinite integral represents a family of functions.
A definite integral has specific limits of integration and represents the area under a curve between those limits. An indefinite integral has no limits and represents a general function that can be differentiated to get the original function.
No, not all functions have an indefinite integral. Some functions, such as exponential and trigonometric functions, have indefinite integrals, while others, like the gamma function, do not have an indefinite integral.
Finding indefinite integrals is used in many fields of mathematics and science, such as physics, engineering, and economics. It is used to solve various real-world problems, including finding areas and volumes, determining work and energy, and modeling rates of change.