int(6x^(2))/(sqrt(3x+5))dx

Identify integral: Identify the integral to be solved. We have the integral: int(6x^2)/(sqrt(3x+5))dx We need to find a way to simplify this integral to make it easier to solve.Use substitution: Use substitution to simplify the integral. Let u = 3x + 5. Then, du/dx = 3, or du = 3dx. We need to express x^2 and dx in terms of u and du.Solve for x: Solve for x in terms of u. Since u = 3x + 5, we have x = (u - 5)/3. Now we can find x^2 in terms of u. x^2 = ((u - 5)/3)^2Substitute x^2 and dx: Substitute x^2 and dx in the integral. x^2 = ((u - 5)/3)^2 dx = du/3 Now, substitute these into the integral: int(6((u - 5)/3)^2)/(sqrt(u)) * (du/3)Simplify the integral: Simplify the integral. int(6((u - 5)/3)^2)/(sqrt(u)) * (du/3) = int(6(u^2 - 10u + 25)/9)/(sqrt(u)) * (du/3) = int(2(u^2 - 10u + 25)/9)/(sqrt(u)) * du = (2/9)int(u^2 - 10u + 25)/(sqrt(u)) duSplit into integrals: Split the integral into three separate integrals. (2/9)int(u^2 - 10u + 25)/(sqrt(u)) du = (2/9)(int(u^(3/2) du) - 10int(u^(1/2) du) + 25int(u^(-1/2) du))Integrate each term: Integrate each term separately. int(u^(3/2) du) = (2/5)u^(5/2) + C1 int(u^(1/2) du) = (2/3)u^(3/2) + C2 int(u^(-1/2) du) = 2u^(1/2) + C3Combine integrals: Combine the integrals and multiply by the constant. (2/9)((2/5)u^(5/2) - 10(2/3)u^(3/2) + 25(2)u^(1/2)) + C = (2/9)((4/5)u^(5/2) - (20/3)u^(3/2) + 50u^(1/2)) + C = (8/45)u^(5/2) - (40/27)u^(3/2) + (100/9)u^(1/2) + CSubstitute back for u: Substitute back for u to get the integral in terms of x. u = 3x + 5 (8/45)(3x + 5)^(5/2) - (40/27)(3x + 5)^(3/2) + (100/9)(3x + 5)^(1/2) + C

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$\int\frac{6x^{2}}{\sqrt{3x+5}}dx$

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Evaluate definite integrals using the chain rule

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\int\frac{6x^{2}}{\sqrt{3x+5}}dx

Identify integral: Identify the integral to be solved. $\newline$ We have the integral: $\newline$ $\int\frac{6x^2}{\sqrt{3x+5}}\,dx$ $\newline$ We need to find a way to simplify this integral to make it easier to solve.
Use substitution: Use substitution to simplify the integral. $\newline$ Let $u = 3x + 5$ . Then, $\frac{du}{dx} = 3$ , or $du = 3dx$ . We need to express $x^2$ and $dx$ in terms of $u$ and $du$ .
Solve for x: Solve for x in terms of $u$ . $\newline$ Since $u = 3x + 5$ , we have $x = \frac{u - 5}{3}$ . Now we can find $x^2$ in terms of $u$ . $\newline$ $x^2 = \left(\frac{u - 5}{3}\right)^2$
Substitute $x^2$ and $dx$ : Substitute $x^2$ and $dx$ in the integral. $\newline$ $x^2 = \left(\frac{u - 5}{3}\right)^2$ $\newline$ $dx = \frac{du}{3}$ $\newline$ Now, substitute these into the integral: $\newline$ $\int \frac{6\left(\frac{u - 5}{3}\right)^2}{\sqrt{u}} \cdot \left(\frac{du}{3}\right)$
Simplify the integral: Simplify the integral. $\newline$ $\int\frac{6\left(\frac{u - 5}{3}\right)^2}{\sqrt{u}} \cdot \frac{du}{3}$ $\newline$ = $\int\frac{6(u^2 - 10u + 25)}{9}\frac{1}{\sqrt{u}} \cdot \frac{du}{3}$ $\newline$ = $\int\frac{2(u^2 - 10u + 25)}{9}\frac{1}{\sqrt{u}} \cdot du$ $\newline$ = $\frac{2}{9}\int\frac{u^2 - 10u + 25}{\sqrt{u}} du$
Split into integrals: Split the integral into three separate integrals. $\newline$ $(\frac{2}{9})\int(u^2 - 10u + 25)/(\sqrt{u}) \, du$ $\newline$ = $(\frac{2}{9})(\int u^{\frac{3}{2}} \, du) - 10\int u^{\frac{1}{2}} \, du + 25\int u^{-\frac{1}{2}} \, du)$
Integrate each term: Integrate each term separately. $\newline$ $\int u^{\frac{3}{2}} \, du = \frac{2}{5}u^{\frac{5}{2}} + C_1$ $\newline$ $\int u^{\frac{1}{2}} \, du = \frac{2}{3}u^{\frac{3}{2}} + C_2$ $\newline$ $\int u^{-\frac{1}{2}} \, du = 2u^{\frac{1}{2}} + C_3$
Combine integrals: Combine the integrals and multiply by the constant. $\newline$ $(\frac{2}{9})((\frac{2}{5})u^{\frac{5}{2}} - 10(\frac{2}{3})u^{\frac{3}{2}} + 25(2)u^{\frac{1}{2}}) + C$ $\newline$ = $(\frac{2}{9})((\frac{4}{5})u^{\frac{5}{2}} - (\frac{20}{3})u^{\frac{3}{2}} + 50u^{\frac{1}{2}}) + C$ $\newline$ = $(\frac{8}{45})u^{\frac{5}{2}} - (\frac{40}{27})u^{\frac{3}{2}} + (\frac{100}{9})u^{\frac{1}{2}} + C$
Substitute back for u: Substitute back for u to get the integral in terms of x. $\newline$ $u = 3x + 5$ $\newline$ $\frac{8}{45}(3x + 5)^{\frac{5}{2}} - \frac{40}{27}(3x + 5)^{\frac{3}{2}} + \frac{100}{9}(3x + 5)^{\frac{1}{2}} + C$