The antiderivative of sqrtx represents a foundational concept in integral calculus, useful for modeling area under curves and solving differential equations. Finding this antiderivative involves reversing the power rule and adjusting for the constant of integration.
Mastering the integral of square root of x builds confidence for more advanced applications in physics, engineering, and data analysis. This article explains the process step by step with clarity and practical context.
| Function | Power Form | Antiderivative | Check by Derivative |
|---|---|---|---|
| sqrtx | x^(1/2) | (2/3) x^(3/2) + C | (3/2) * (2/3) x^(1/2) = sqrtx |
| 1/sqrtx | x^(-1/2) | 2 x^(1/2) + C | -2 * (-1/2) x^(-3/2) = x^(-1/2) |
| sqrt(ax) | (ax)^(1/2) | (2/(3a)) (ax)^(3/2) + C | Derivative returns sqrt(ax) |
| sqrt(x^2 + k) | (x^2 + k)^(1/2) | Requires trigonometric or hyperbolic substitution | Verification by chain rule |
Rewrite sqrtx as a Power Function
Before integrating, rewrite the square root in exponent form. The expression sqrtx is equivalent to x raised to the one-half power, which aligns with the standard power rule for integration.
Expressing the function this way clarifies how to apply the rule and reduces algebraic mistakes in later steps.
Apply the Power Rule for Integration
Increase the exponent by one, then divide by the new exponent. For x^(1/2), adding one gives x^(3/2), and dividing by 3/2 is the same as multiplying by 2/3.
This straightforward step produces the core structure of the antiderivative, with the constant of integration added to account for all possible vertical shifts.
Verification by Differentiation
Differentiate (2/3) x^(3/2) + C to confirm the result. The power rule for derivatives brings down 3/2, which cancels with the coefficient 2/3, leaving sqrtx as the original function.
Consistent verification strengthens understanding and builds reliability when solving more complex integrals.
Handling Coefficients and Transformations
When the expression involves coefficients inside the radical, such as sqrt(ax), adjust the power rule accordingly. Factor constants so that integration follows the same logic as the basic sqrtx case.
Recognizing these patterns ensures accurate results and smoother problem-solving in applied contexts.
Key Takeaways for the Antiderivative of sqrtx
- Rewrite sqrtx as x^(1/2) to apply the power rule.
- Increase the exponent by one and divide by the new exponent, yielding (2/3) x^(3/2).
- Add the constant of integration C to represent the family of antiderivatives.
- Verify by differentiating to ensure the result matches the original function.
- Extend the method to coefficients and transformed arguments with careful algebraic adjustment.
FAQ
Reader questions
How do I find the antiderivative of sqrtx using substitution?
Set u = x so that du = dx, and apply the power rule directly to u^(1/2). The substitution confirms the standard formula and is helpful when sqrtx appears inside more complicated integrands.
What is the antiderivative of sqrtx if a constant multiplies the variable?
For a constant a, integrate sqrt(ax) by rewriting as (ax)^(1/2) and adjusting for the chain rule. The result is (2/(3a)) (ax)^(3/2) + C, which can be verified by differentiation. Yes, evaluate (2/3) x^(3/2) at the upper and lower bounds and subtract. This method works for any finite interval where sqrtx is defined and continuous. Simplify sqrt(x^2) to |x| before integrating, and treat the absolute value carefully across different domains. For positive x, the antiderivative matches the standard power rule result.