QuestionDecember 13, 2025

int _(1)^41tsqrt (16+33t)dt=square

int _(1)^41tsqrt (16+33t)dt=square
int _(1)^41tsqrt (16+33t)dt=square

Solution
4.0(130 votes)

Answer

25898.25 Explanation 1. Use substitution Let u = 16 + 33t, so du = 33\,dt and t = \frac{u - 16}{33}. When t = 1, u = 49; when t = 41, u = 1379. Thus, \int_{1}^{41} t\sqrt{16 + 33t} \, dt = \int_{49}^{1379} \frac{u - 16}{33} \cdot u^{1/2} \cdot \frac{du}{33}. 2. Simplify the integral Factor constants: \frac{1}{33^2} \int_{49}^{1379} (u - 16) u^{1/2} \, du. Expand: \frac{1}{33^2} \int_{49}^{1379} \left( u^{3/2} - 16u^{1/2} \right) du. 3. Integrate **\int u^{3/2} du = \frac{2}{5} u^{5/2}**, **\int u^{1/2} du = \frac{2}{3} u^{3/2}**. So: \frac{1}{33^2} \left[ \frac{2}{5}u^{5/2} - \frac{32}{3}u^{3/2} \right]_{49}^{1379}. 4. Evaluate at bounds For u = 1379: u^{1/2} = 37.128, u^{3/2} \approx 51183.9, u^{5/2} \approx 70510995.8. For u = 49: u^{1/2} = 7, u^{3/2} = 343, u^{5/2} = 16807. Plug in: Top = \frac{2}{5}(70510995.8) - \frac{32}{3}(51183.9). Bottom = \frac{2}{5}(16807) - \frac{32}{3}(343). Difference \approx 28204398.32 - 6722.1333 \approx 28197676.187. 5. Multiply constant Final: \frac{28197676.187}{1089} \approx 25898.252.

Explanation

1. Use substitution<br /> Let $u = 16 + 33t$, so $du = 33\,dt$ and $t = \frac{u - 16}{33}$. <br />When $t = 1$, $u = 49$; when $t = 41$, $u = 1379$. <br />Thus, <br />$\int_{1}^{41} t\sqrt{16 + 33t} \, dt = \int_{49}^{1379} \frac{u - 16}{33} \cdot u^{1/2} \cdot \frac{du}{33}$.<br /><br />2. Simplify the integral<br /> Factor constants: <br />$\frac{1}{33^2} \int_{49}^{1379} (u - 16) u^{1/2} \, du$. <br />Expand: $\frac{1}{33^2} \int_{49}^{1379} \left( u^{3/2} - 16u^{1/2} \right) du$.<br /><br />3. Integrate<br /> **$\int u^{3/2} du = \frac{2}{5} u^{5/2}$**, <br />**$\int u^{1/2} du = \frac{2}{3} u^{3/2}$**. <br />So: <br />$\frac{1}{33^2} \left[ \frac{2}{5}u^{5/2} - \frac{32}{3}u^{3/2} \right]_{49}^{1379}$.<br /><br />4. Evaluate at bounds<br /> For $u = 1379$: <br />$u^{1/2} = 37.128$, $u^{3/2} \approx 51183.9$, $u^{5/2} \approx 70510995.8$. <br />For $u = 49$: <br />$u^{1/2} = 7$, $u^{3/2} = 343$, $u^{5/2} = 16807$. <br />Plug in: <br />Top = $\frac{2}{5}(70510995.8) - \frac{32}{3}(51183.9)$. <br />Bottom = $\frac{2}{5}(16807) - \frac{32}{3}(343)$.<br /><br />Difference $\approx 28204398.32 - 6722.1333 \approx 28197676.187$.<br /><br />5. Multiply constant<br /> Final: $\frac{28197676.187}{1089} \approx 25898.252$.
Click to rate:

Similar Questions