I'm taking this to meanNawz said:Homework Statement
Evaluate:
Integral sign: Square root of 3x times dx
Nawz said:Homework Equations
The Attempt at a Solution
integral of 3x^1/2 dx
(3x^(3/2)) / (3/2)
so I multiplied it by 2/3 and got:
(2/3x^(3/2)) +C
but the answer is 2 times the square root of 3 divided by 3 then x^3/2.. ? I don't know how they kept the square root of 3.
Mark44 said:I'm taking this to mean
[tex]\int \sqrt{3x}dx[/tex]
The simplest way to approach this is to bring out a factor of sqrt(3), and rewriting the integral using exponents, rather than radicals.
[tex]\sqrt{3}\int x^{1/2}~dx[/tex]
Can you carry out the rest of this?
The purpose of evaluating integrals in calculus is to find the area under a curve and solve problems related to accumulation, such as finding distance traveled or volume of a shape.
To solve an integral problem, you must first identify the function to be integrated and the limits of integration. Then, you can use various integration techniques, such as substitution, integration by parts, or partial fraction decomposition, to find the antiderivative. Finally, you can evaluate the integral by plugging in the limits of integration and simplifying.
A definite integral has specific limits of integration and gives a numerical value as the result, representing the area under the curve. An indefinite integral does not have limits and gives a function as the result, representing the antiderivative of the original function.
Some common mistakes to avoid when evaluating integrals include forgetting to include the constant of integration, incorrect use of integration techniques, and using the wrong limits of integration. It is also important to double-check your work and simplify your answer if possible.
Understanding integrals can benefit you outside of calculus by helping you solve real-world problems involving accumulation, such as finding the average rate of change or the total cost of a product. It can also improve your overall understanding of mathematical concepts and critical thinking skills.