Solve \int x^(5) 1/x^(2) root(3)(x) sqrt(x)

Question

Evaluate the integral

\frac{6}{29} x^{4} 6 x^{5} + C, C \in R

Evaluate

\int x^{5} \times \frac{1}{x ^{2}} \times 3 x \times x d x

Evaluate the power

\int x^{5} \times \frac{1}{x ^{2}} \times x^{\frac{1}{3}} x d x

Multiply

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\int x^{3} 6 x^{5} d x

Prepare for integration by parts

u = x^{3} d v = 6 x^{5} d x

Calculate the derivative

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d u = 3 x^{2} d x d v = 6 x^{5} d x

Evaluate the integral

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d u = 3 x^{2} d x v = \frac{6}{11} x^{\frac{11}{6}}

Substitute u = x^{3} 、 v = \frac{6}{11} x^{\frac{11}{6}} 、 d u = 3 x^{2} d x 、 d v = 6 x^{5} d x for \int u d v = uv - \int v d u

x^{3} \times \frac{6}{11} x^{\frac{11}{6}} - \int 3 x^{2} \times \frac{6}{11} x^{\frac{11}{6}} d x

Calculate

\frac{6}{11} x^{\frac{29}{6}} - \int \frac{18}{11} x^{\frac{23}{6}} d x

Evaluate the integral

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\frac{6}{11} x^{\frac{29}{6}} - \frac{108}{319} x^{\frac{29}{6}}

Collect like terms by calculating the sum or difference of their coefficients

(\frac{6}{11} - \frac{108}{319}) x^{\frac{29}{6}}

Subtract the numbers

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\frac{6}{29} x^{\frac{29}{6}}

Use a^{\frac{m}{n}} = n a^{m} to transform the expression

\frac{6}{29} 6 x^{29}

Simplify the radical expression

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\frac{6}{29} x^{4} 6 x^{5}

Solution

\frac{6}{29} x^{4} 6 x^{5} + C, C \in R

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