Solve \int \sqrt x \sqrt x^22 d x

Question

Evaluate the integral

\frac{4}{5} x^{2} x + C, C \in R

Evaluate

\int x \times (x)^{2} \times 2 d x

Multiply

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\int 2 x x d x

Use the property of integral \int k f (x) d x = k \int f (x) d x

2 \times \int x x d x

Prepare for integration by parts

u = x d v = x d x

Calculate the derivative

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d u = d x d v = x d x

Evaluate the integral

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

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

2 (x \times \frac{2}{3} x^{\frac{3}{2}} - \int 1 \times \frac{2}{3} x^{\frac{3}{2}} d x)

Calculate

2 (\frac{2}{3} x^{\frac{5}{2}} - \int \frac{2}{3} x^{\frac{3}{2}} d x)

Calculate

\frac{4}{3} x^{\frac{5}{2}} - 2 \times \int \frac{2}{3} x^{\frac{3}{2}} d x

Evaluate the integral

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\frac{4}{3} x^{\frac{5}{2}} - \frac{8}{15} x^{\frac{5}{2}}

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

(\frac{4}{3} - \frac{8}{15}) x^{\frac{5}{2}}

Subtract the numbers

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\frac{4}{5} x^{\frac{5}{2}}

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

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

Simplify the radical expression

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\frac{4}{5} x^{2} x

Solution

\frac{4}{5} x^{2} x + C, C \in R

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