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Homework Statement
Antiderivative of sqrt[e^x+ln(x^2)+1]
Homework Equations
The Attempt at a Solution
(2[(e^x+ln(x^2)+1)^(3/2)]x^2)/[3(e^x+2x)]
The antiderivative of square root is a function that, when differentiated, gives the square root function as its result. It is the inverse operation of differentiation.
To find the antiderivative of square root, you can use the power rule of integration, which states that the antiderivative of x^n is (x^(n+1))/(n+1). Therefore, the antiderivative of √x is (x^(1/2))/(1/2+1) = (2x^(1/2))/3 + C, where C is the constant of integration.
The derivative of square root is a function that gives the rate of change of the square root function at a particular point, while the antiderivative of square root is a function that gives the original function (square root) when differentiated. In other words, the derivative and antiderivative are inverse operations.
Yes, you can use the chain rule to find the antiderivative of square root. For example, if you have the integral ∫√(x^2+1) dx, you can use u-substitution and let u = x^2+1. Then, du/dx = 2x, and dx = du/2x. Substituting back into the integral gives ∫√u * (du/2x) = (1/2)∫√u * (1/x) du. Using the power rule, the antiderivative is (1/2) * (2/3 * u^(3/2)) * (1/x) + C = (1/3)*x*(x^2+1)^(3/2) + C.
Understanding the antiderivative of square root is important in many areas of mathematics and science, as it is a fundamental concept in calculus. It allows us to solve problems involving rates of change, optimization, and motion, among other applications. Additionally, it forms the basis for more advanced integration techniques and is crucial in understanding the behavior of functions and their derivatives.