What is the integral of (3x^3+1)^2\sqrt[3]{x^7} ?

The integral of (3x^3+1)^2\sqrt[3]{x^7} is 27/28 x^{28/3}+18/19 x^{19/3}+3/10 x^{10/3}+C

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integral of (3x^3+1)^2\sqrt[3]{x^7}

\int (3 x^{3} + 1)^{2} 3 x^{7} d x

\frac{27}{28} x^{\frac{28}{3}} + \frac{18}{19} x^{\frac{19}{3}} + \frac{3}{10} x^{\frac{10}{3}} + C

Solution steps

\int (3 x^{3} + 1)^{2} 3 x^{7} d x

Apply u-substitution

: \int \frac{3 u ^{\frac{3}{7}} ( 3 u ^{\frac{9}{7}} + 1 ) ^{2}}{7} d u

= \int \frac{3 u ^{\frac{3}{7}} ( 3 u ^{\frac{9}{7}} + 1 ) ^{2}}{7} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{3}{7} \cdot \int u^{\frac{3}{7}} (3 u^{\frac{9}{7}} + 1)^{2} d u

Expand

u^{\frac{3}{7}} (3 u^{\frac{9}{7}} + 1)^{2} : 9 u^{3} + 6 u^{\frac{12}{7}} + u^{\frac{3}{7}}

= \frac{3}{7} \cdot \int 9 u^{3} + 6 u^{\frac{12}{7}} + u^{\frac{3}{7}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{3}{7} (\int 9 u^{3} d u + \int 6 u^{\frac{12}{7}} d u + \int u^{\frac{3}{7}} d u)

\int 9 u^{3} d u = \frac{9 u ^{4}}{4}

\int 6 u^{\frac{12}{7}} d u = \frac{42}{19} u^{\frac{19}{7}}

\int u^{\frac{3}{7}} d u = \frac{7}{10} u^{\frac{10}{7}}

= \frac{3}{7} (\frac{9 u ^{4}}{4} + \frac{42}{19} u^{\frac{19}{7}} + \frac{7}{10} u^{\frac{10}{7}})

Substitute back

u = 3 x^{7}

= \frac{3}{7} \frac{9 ( 3 x ^{7} ) ^{4}}{4} + \frac{42}{19} (3 x^{7})^{\frac{19}{7}} + \frac{7}{10} (3 x^{7})^{\frac{10}{7}}

Simplify

\frac{3}{7} \frac{9 ( 3 x ^{7} ) ^{4}}{4} + \frac{42}{19} (3 x^{7})^{\frac{19}{7}} + \frac{7}{10} (3 x^{7})^{\frac{10}{7}} : \frac{27}{28} x^{\frac{28}{3}} + \frac{18}{19} x^{\frac{19}{3}} + \frac{3}{10} x^{\frac{10}{3}}

= \frac{27}{28} x^{\frac{28}{3}} + \frac{18}{19} x^{\frac{19}{3}} + \frac{3}{10} x^{\frac{10}{3}}

Add a constant to the solution

= \frac{27}{28} x^{\frac{28}{3}} + \frac{18}{19} x^{\frac{19}{3}} + \frac{3}{10} x^{\frac{10}{3}} + C

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What is the integral of (3x^3+1)^2\sqrt[3]{x^7} ?
The integral of (3x^3+1)^2\sqrt[3]{x^7} is 27/28 x^{28/3}+18/19 x^{19/3}+3/10 x^{10/3}+C