Why choose $u = 1 + 9/4x$ in this arc-length integral?

Conditions

• The integral to evaluate is $\int_0^{32/9}\sqrt{1+\frac{9}{4}x}\,dx$.
• The integrand contains a composite expression under a square root.

Reasoning, step by step

1. Identify the expression under the square root in the integrand, which is $1 + \frac{9}{4}x$.
2. Set the substitution variable $u$ equal to this expression: $u = 1 + \frac{9}{4}x$.
3. Differentiate $u$ with respect to $x$ to find $\frac{du}{dx} = \frac{9}{4}$.
4. Solve for $dx$ to get $dx = \frac{4}{9}du$.
5. Substitute $u$ and $dx$ into the original integral to simplify it to $\frac{4}{9}\int \sqrt{u}\,du$.

Example

The instructor explicitly states, "Straight up use substitution. So u substitution," and writes $u = 1 + \frac{9}{4}x$ because it is exactly the expression under the radical, turning the integrand into $\sqrt{u}$.

Common misconceptions

• Choosing a substitution that does not simplify the integrand, such as setting $u$ to just $x$.
• Forgetting to change the differential $dx$ to $du$ when substituting.
• Assuming the substitution works without adjusting the limits of integration for a definite integral.

Watch the explanation

Explore next

• How do the limits change when using u-substitution in a definite integral?
• What is the antiderivative of $\sqrt{u}$ used here?