What is the antiderivative of $\sqrt{u}$ used here?

Conditions

• The integrand is $\sqrt{u}$, which is equivalent to $u^{1/2}$.
• The power rule for integration is applicable.
• $u \ge 0$ for the real-valued square root form.

Reasoning, step by step

1. Rewrite the integrand $\sqrt{u}$ as $u^{1/2}$.
2. Apply the power rule for integration: add 1 to the exponent to get $3/2$.
3. Divide by the new exponent $3/2$.
4. Simplify the expression $\frac{u^{3/2}}{3/2}$ by multiplying by its reciprocal, resulting in $\frac{2}{3}u^{3/2}$.

Example

The instructor explains that the antiderivative of $u^{1/2}$ is $u^{3/2}$ divided by $3/2$, which is multiplying by $2/3$. The board shows the antiderivative as $\frac{2}{3}u^{3/2}$.

Common misconceptions

• Forgetting to add 1 to the exponent when applying the power rule.
• Incorrectly dividing by the original exponent instead of the new exponent.
• Failing to simplify the complex fraction $\frac{1}{3/2}$ to $\frac{2}{3}$.

Watch the explanation

Explore next

• How do you evaluate the definite integral $\frac{4}{9}\int_{1}^{9}\sqrt{u}$\,du?
• Why does $9^{3/2}-1^{3/2}$ become 26 in this worked example?