Calculate The Indefinite Integral of Y Square Root of X

What is an indefinite integral?

An indefinite integral represents the antiderivative of a function. It finds all functions whose derivative equals the original function. The result includes a constant of integration (C) because derivatives of constants are zero.

For a function f(x), the indefinite integral is written as ∫f(x) dx. The notation means "the antiderivative of f(x) with respect to x."

Integral of √x (∫√x dx)

The integral of the square root of x is a fundamental calculus problem. The square root function can be written as x^(1/2), so we'll use power rule integration.

The result is (2/3)x^(3/2) + C, where C is the constant of integration. This means the antiderivative of √x is (2/3)x^(3/2) plus any constant.

Step-by-step solution

1. Rewrite √x as x^(1/2)
2. Apply the power rule for integration: ∫x^n dx = (x^(n+1))/(n+1) + C
3. Substitute n = 1/2: ∫x^(1/2) dx = (x^(1/2 + 1))/(1/2 + 1) + C
4. Simplify the exponents: (x^(3/2))/(3/2) + C
5. Simplify the fraction: (2/3)x^(3/2) + C

Example calculation

Let's find ∫√x dx from x=1 to x=4.

First, find the antiderivative: (2/3)x^(3/2) + C

Evaluate at the bounds:

• At x=4: (2/3)(4)^(3/2) = (2/3)(8) = 16/3
• At x=1: (2/3)(1)^(3/2) = (2/3)(1) = 2/3

Subtract the lower bound from the upper bound: (16/3) - (2/3) = 14/3 ≈ 4.6667

Common mistakes

• Forgetting the constant of integration (C)
• Incorrectly applying the power rule (adding exponents instead of adding 1)
• Miscounting the exponent when simplifying (3/2 instead of 1/2 + 1)
• Not simplifying the final fraction (leaving it as x^(3/2)/(3/2))