Calculate the integral and write your answer in simplest form. int(5root(4)(x^(5)))/(6)dx Answer:

Rewrite in Exponents: We are given the integral to solve: \[ \int \frac{5\sqrt[4]{x^5}}{6} \, dx \] First, we rewrite the integral in terms of exponents to simplify the integration process. \[ \sqrt[4]{x^5} = x^{5/4} \] So the integral becomes: \[ \int \frac{5x^{5/4}}{6} \, dx \]Pull Out Constant Factor: Now we can pull out the constant factor from the integral: \[ \frac{5}{6} \int x^{5/4} \, dx \]Apply Power Rule: 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 $ and $ C $ is the constant of integration. For our integral, $ n = 5/4 $, so we get: \[ \frac{5}{6} \int x^{5/4} \, dx = \frac{5}{6} \cdot \frac{x^{(5/4)+1}}{(5/4)+1} + C \]Simplify Exponent and Fraction: We simplify the exponent and the fraction: \[ (5/4) + 1 = 5/4 + 4/4 = 9/4 \] So the integral becomes: \[ \frac{5}{6} \cdot \frac{x^{9/4}}{9/4} + C \]Multiply by Reciprocal: To simplify the fraction, we multiply by the reciprocal of $ 9/4 $: \[ \frac{5}{6} \cdot \frac{4}{9} \cdot x^{9/4} + C \]Multiply Constants: Now we multiply the constants: \[ \frac{5 \cdot 4}{6 \cdot 9} \cdot x^{9/4} + C = \frac{20}{54} \cdot x^{9/4} + C \]Simplify Fraction: We can simplify the fraction $ \frac{20}{54} $ by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \[ \frac{20}{54} = \frac{10}{27} \] So the integral simplifies to: \[ \frac{10}{27} \cdot x^{9/4} + C \]

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Calculate the integral and write your answer in simplest form.

int(5root(4)(x^(5)))/(6)dx
Answer:

Calculate the integral and write your answer in simplest form. $\newline$ $\int \frac{5 \sqrt[4]{x^{5}}}{6} \mathrm{dx}$ $\newline$ Answer:

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Calculus

Evaluate definite integrals using the power rule

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Q. Calculate the integral and write your answer in simplest form.

\newline

\int \frac{5 \sqrt[4]{x^{5}}}{6} \mathrm{dx}

\newline

Answer:

Rewrite in Exponents: We are given the integral to solve: $\newline$ $\int \frac{5\sqrt[4]{x^5}}{6} \, dx$ $\newline$ First, we rewrite the integral in terms of exponents to simplify the integration process. $\newline$ $\sqrt[4]{x^5} = x^{5/4}$ $\newline$ So the integral becomes: $\newline$ $\int \frac{5x^{5/4}}{6} \, dx$
Pull Out Constant Factor: Now we can pull out the constant factor from the integral: $\newline$ $\frac{5}{6} \int x^{5/4} \, dx$
Apply Power Rule: Next, we apply the power rule for integration, which states that: $\newline$ $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ $\newline$ where $n \neq -1$ and $C$ is the constant of integration. For our integral, $n = 5/4$ , so we get: $\newline$ $\frac{5}{6} \int x^{5/4} \, dx = \frac{5}{6} \cdot \frac{x^{(5/4)+1}}{(5/4)+1} + C$
Simplify Exponent and Fraction: We simplify the exponent and the fraction: $\newline$ $(5/4) + 1 = 5/4 + 4/4 = 9/4$ $\newline$ So the integral becomes: $\newline$ $\frac{5}{6} \cdot \frac{x^{9/4}}{9/4} + C$
Multiply by Reciprocal: To simplify the fraction, we multiply by the reciprocal of $9/4$ : $\newline$ $\frac{5}{6} \cdot \frac{4}{9} \cdot x^{9/4} + C$
Multiply Constants: Now we multiply the constants: $\newline$ $\frac{5 \cdot 4}{6 \cdot 9} \cdot x^{9/4} + C = \frac{20}{54} \cdot x^{9/4} + C$
Simplify Fraction: We can simplify the fraction $\frac{20}{54}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $2$ : $\newline$ $\frac{20}{54} = \frac{10}{27}$ $\newline$ So the integral simplifies to: $\newline$ $\frac{10}{27} \cdot x^{9/4} + C$