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When using u-substitution in a definite integral, the original limits of integration in terms of $x$ must be converted into new limits in terms of $u$. This is done by substituting the original $x$-bounds into the substitution equation $u(x)$.

Conditions: The integral is a definite integral.; A substitution $u = u(x)$ is being used.; The original limits of integration are given in terms of $x$.

When using u-substitution in a definite integral, the original limits of integration in terms of $x$ must be converted into new limits in terms of $u$. This is done by substituting the original $x$-bounds into the substitution equation $u(x)$.

Conditions: The integral is a definite integral.; A substitution $u = u(x)$ is being used.; The original limits of integration are given in terms of $x$.

The substitution $u = 1 + \frac{9}{4}x$ is chosen because it is exactly the expression under the radical in the arc-length integral $\int_0^{32/9}\sqrt{1+\frac{9}{4}x}\,dx$. By setting $u$ to this inner expression, the complicated integrand simplifies to $\sqrt{u}$, which is much easier to integrate using the power rule.

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.

The substitution $u = 1 + \frac{9}{4}x$ is chosen because it is exactly the expression under the radical in the arc-length integral $\int_0^{32/9}\sqrt{1+\frac{9}{4}x}\,dx$. By setting $u$ to this inner expression, the complicated integrand simplifies to $\sqrt{u}$, which is much easier to integrate using the power rule.

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.