Resolved: "Give the correct answer and solve the given..."

Emily-Jane Bray

Answered question

2020-11-11

Give the correct answer and solve the given equation Evaluate

\int x^{3} (\sqrt[3]{1 - x^{2}}) d x

l1koV

Skilled2020-11-12Added 100 answers

We have to find

\int x^{3} (\sqrt[3]{1 - x^{2}}) d x

Aplly substitution

u = 1 - x^{2}

gives us

d u = - 2 x d x and x = (1 - u)

\int x^{2} \sqrt[3]{1 - x^{2}} d x = \int x^{2} \sqrt[3]{1 - x^{2}} (x d x)

= \int - \frac{(1 - u) \sqrt[2]{u}}{2} d u

= \frac{1}{2} \cdot \int (1 - u) \sqrt[3]{u} d u

= \frac{1}{2} \cdot \int \sqrt[3]{u} - u^{\frac{4}{3}} d u

= - \frac{1}{2} (\int \sqrt[3]{u} d u - \int u^{\frac{4}{3}} d u) [Apply the sum rule of integration]

= - \frac{1}{2} (\frac{u^{\frac{1}{3} + 1}}{\frac{1}{3} + 1} - \frac{u^{\frac{4}{3} + 1}}{\frac{4}{3} + 1}) [∴ x^{n} d x = \frac{x^{n + 1}}{n + 1}, n \neq - 1]

= - \frac{1}{2} (\frac{3}{4} u^{\frac{4}{3}} - \frac{3}{7} u^{\frac{7}{3}})

= - \frac{1}{2} (\frac{3}{4} \frac{{(1 - x^{2})}^{4}}{3} - \frac{3}{7} \frac{{(1 - x^{2})}^{4}}{3}) [∴ u = 1 - x^{2}]

= \frac{3 \frac{{(1 - x^{2})}^{7}}{3}}{14} - \frac{3 \frac{{(1 - x^{2})}^{4}}{3}}{8}

= C - {(1 - x^{2})}^{\frac{4}{3}} \frac{12 x^{2} + 9}{56}

where C is an integrating constant.
The finally answer is

\int x^{3} (\sqrt[3]{1 - x^{2}}) d x = C - {(1 - x^{2})}^{\frac{4}{}}

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