Calculate The Indefinite Integral of Y Square Root of X

Calculating the indefinite integral of y times the square root of x is a common calculus problem. This guide explains the process step-by-step, provides a calculator for quick results, and includes practical examples to help you understand the concept.

What is the indefinite integral of y√x?

The indefinite integral of y times the square root of x, written as ∫y√x dx, represents the antiderivative of the function y√x. This means we're looking for a function F(x) whose derivative is y√x. The result will include an arbitrary constant C because indefinite integrals have infinitely many solutions that differ by a constant.

In calculus, the indefinite integral is also called an antiderivative or primitive function. It's the reverse operation of differentiation.

The general form of the solution depends on whether y is a constant or a function of x. In this guide, we'll assume y is a constant multiplier.

How to calculate the indefinite integral of y√x

To find ∫y√x dx, we can use integration techniques. Here's the step-by-step process:

Rewrite the integrand: √x = x^(1/2)
Factor out the constant y: yx^(1/2)
Use the power rule for integration: ∫x^n dx = x^(n+1)/(n+1) + C, when n ≠ -1
Apply the power rule to yx^(1/2): y * x^(3/2)/(3/2) + C
Simplify the expression: (2y/3)x^(3/2) + C

∫y√x dx = (2y/3)x^(3/2) + C

This formula gives the general solution for the indefinite integral of y times the square root of x.

Worked example

Let's calculate ∫2√x dx using our formula.

Identify y = 2
Apply the formula: (2*2/3)x^(3/2) + C = (4/3)x^(3/2) + C
Simplify: (4/3)√(x³) + C

The result (4/3)x^(3/2) + C is the antiderivative of 2√x. The constant C represents all possible constants that could be added to this function to satisfy the original equation.

To verify this result, you can differentiate (4/3)x^(3/2) + C and confirm that you get back to 2√x.

FAQ

What if y is not a constant?

If y is a function of x, you would need to use integration techniques like substitution or integration by parts, depending on the specific form of y.

Can I calculate this integral with a calculator?

Yes, the calculator on this page can compute the integral for any constant y and x values you provide.

What's the difference between definite and indefinite integrals?

An indefinite integral finds a general family of functions (including an arbitrary constant), while a definite integral calculates a specific area or value between two points.

How do I know when to use this formula?

Use this formula when you have a function that's a constant multiplied by the square root of x, and you need to find its antiderivative.