QuestionAugust 25, 2025

(3 points) Evaluate the definite integral: int _(0)^74xsqrt (x+5)dx=square

(3 points) Evaluate the definite integral: int _(0)^74xsqrt (x+5)dx=square
(3 points) Evaluate the definite integral: int _(0)^74xsqrt (x+5)dx=square

Solution
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Answer

728 Explanation 1. Use Substitution Let u = x + 5, then du = dx and x = u - 5. Change limits: when x = 0, u = 5; when x = 7, u = 12. 2. Substitute and Simplify The integral becomes \int_{5}^{12} 4(u-5)\sqrt{u} \, du = \int_{5}^{12} 4(u^{3/2} - 5u^{1/2}) \, du. 3. Integrate Integrate term by term: \int 4u^{3/2} \, du = \frac{8}{5}u^{5/2} and \int 4 \cdot 5u^{1/2} \, du = \frac{20}{3}u^{3/2}. 4. Evaluate the Integral Evaluate from 5 to 12: \left[\frac{8}{5}u^{5/2} - \frac{20}{3}u^{3/2}\right]_{5}^{12}. 5. Calculate Values Compute: \left(\frac{8}{5}(12)^{5/2} - \frac{20}{3}(12)^{3/2}\right) - \left(\frac{8}{5}(5)^{5/2} - \frac{20}{3}(5)^{3/2}\right).

Explanation

1. Use Substitution<br /> Let $u = x + 5$, then $du = dx$ and $x = u - 5$. Change limits: when $x = 0$, $u = 5$; when $x = 7$, $u = 12$.<br /><br />2. Substitute and Simplify<br /> The integral becomes $\int_{5}^{12} 4(u-5)\sqrt{u} \, du = \int_{5}^{12} 4(u^{3/2} - 5u^{1/2}) \, du$.<br /><br />3. Integrate<br /> Integrate term by term: $\int 4u^{3/2} \, du = \frac{8}{5}u^{5/2}$ and $\int 4 \cdot 5u^{1/2} \, du = \frac{20}{3}u^{3/2}$.<br /><br />4. Evaluate the Integral<br /> Evaluate from 5 to 12: $\left[\frac{8}{5}u^{5/2} - \frac{20}{3}u^{3/2}\right]_{5}^{12}$.<br /><br />5. Calculate Values<br /> Compute: $\left(\frac{8}{5}(12)^{5/2} - \frac{20}{3}(12)^{3/2}\right) - \left(\frac{8}{5}(5)^{5/2} - \frac{20}{3}(5)^{3/2}\right)$.
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