Rewrite square root as power: We are given the integral to solve: ∫5√(5x)dx. To solve this, we will first rewrite the square root as a power of 1/2. ∫5(5x)^(1/2)dxPull constant outside integral: Now, we can pull the constant 5 outside the integral to simplify the expression. 5∫(5x)^(1/2)dxApply power rule for integration: Next, we apply the power rule for integration, which states that ∫x^n dx = x^(n+1)/(n+1) + C, where n ≠ -1. In our case, n = 1/2. 5 * [(5x)^(1/2 + 1)] / (1/2 + 1) + CSimplify exponent and denominator: We simplify the exponent and the denominator. 5 * (5x)^(3/2) / (3/2) + CDivide by reciprocal: To divide by 3/2, we multiply by its reciprocal, which is 2/3. (5 * 2/3) * (5x)^(3/2) + CMultiply constants outside integral: Now, we multiply the constants outside the integral. (10/3) * (5x)^(3/2) + CRewrite expression in simplified form: Finally, we can rewrite the expression in a more simplified form. (10/3) * 5^(3/2) * x^(3/2) + CSimplify square root term: Since 5^(3/2) is the square root of 5 cubed, we can simplify it to 5√5. (10/3) * 5√5 * x^(3/2) + C

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$\int 5\sqrt{5x}\,dx$

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Calculus

Find indefinite integrals using the substitution

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\int 5\sqrt{5x}\,dx

Rewrite square root as power: We are given the integral to solve: $\int 5\sqrt{5x}\,dx$ . To solve this, we will first rewrite the square root as a power of $1/2$ . $\newline$ $\int 5(5x)^{1/2}\,dx$
Pull constant outside integral: Now, we can pull the constant $5$ outside the integral to simplify the expression. $5\int (5x)^{\frac{1}{2}}\,dx$
Apply power rule for integration: Next, we apply the power rule for integration, which states that $\int x^n \, dx = \frac{x^{(n+1)}}{(n+1)} + C$ , where $n \neq -1$ . In our case, $n = \frac{1}{2}$ . $5 \times \left[\left(5x\right)^{\left(\frac{1}{2} + 1\right)}\right] / \left(\frac{1}{2} + 1\right) + C$
Simplify exponent and denominator: We simplify the exponent and the denominator. $5 \times (5x)^{\frac{3}{2}} / \left(\frac{3}{2}\right) + C$
Divide by reciprocal: To divide by $\frac{3}{2}$ , we multiply by its reciprocal, which is $\frac{2}{3}$ . $\left(5 \times \frac{2}{3}\right) \times \left(5x\right)^{\frac{3}{2}} + C$
Multiply constants outside integral: Now, we multiply the constants outside the integral. $(\frac{10}{3}) \times (5x)^{\frac{3}{2}} + C$
Rewrite expression in simplified form: Finally, we can rewrite the expression in a more simplified form. $(\frac{10}{3}) \times 5^{(\frac{3}{2})} \times x^{(\frac{3}{2})} + C$
Simplify square root term: Since $5^{\frac{3}{2}}$ is the square root of $5$ cubed, we can simplify it to $5\sqrt{5}$ . $\newline$ $(\frac{10}{3}) \cdot 5\sqrt{5} \cdot x^{\frac{3}{2}} + C$