Now, if $u = x^{3} + 6$ , then $\int x^{2} {(x^{3} + 6)}^{10} d x = \int u^{10} (\frac{1}{3} d u) = \frac{1}{3} \int u^{10} d u$ .
This evaluates as
$\frac{1}{3} \int u^{10} d u = \frac{1}{3} ◻ + c = ◻ + C$

Answer

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Answer

$Final Answer: The integral of x^{2} {(x^{3} + 6)}^{10} with respect to x is \frac{x^{31}}{31} + \frac{15 x^{28}}{7} + \frac{324 x^{25}}{5} + \frac{12960 x^{22}}{11} + \frac{272160 x^{19}}{19} + 122472 x^{16} + \frac{9797760 x^{13}}{13} + 3359232 x^{10} + \frac{75582720 x^{7}}{7} + 25194240 x^{4} + 60466176 x + C$

Steps

Step 1 ：Let $u = x^{3} + 6$ , then $\int x^{2} {(x^{3} + 6)}^{10} d x = \int u^{10} (\frac{1}{3} d u) = \frac{1}{3} \int u^{10} d u$ .

Step 2 ：The integral of $u^{10}$ is $\frac{1}{11} u^{11}$ .

Step 3 ：Substitute $u$ back in to get the final answer: $u = x^{3} + 6$ .

Step 4 ：The integral of $x^{2} {(x^{3} + 6)}^{10}$ with respect to $x$ is $\frac{x^{31}}{31} + \frac{15 x^{28}}{7} + \frac{324 x^{25}}{5} + \frac{12960 x^{22}}{11} + \frac{272160 x^{19}}{19} + 122472 x^{16} + \frac{9797760 x^{13}}{13} + 3359232 x^{10} + \frac{75582720 x^{7}}{7} + 25194240 x^{4} + 60466176 x + C$ .

Step 5 ： $Final Answer: The integral of x^{2} {(x^{3} + 6)}^{10} with respect to x is \frac{x^{31}}{31} + \frac{15 x^{28}}{7} + \frac{324 x^{25}}{5} + \frac{12960 x^{22}}{11} + \frac{272160 x^{19}}{19} + 122472 x^{16} + \frac{9797760 x^{13}}{13} + 3359232 x^{10} + \frac{75582720 x^{7}}{7} + 25194240 x^{4} + 60466176 x + C$

Now, if u=x3+6, then ∫x2(x3+6)10dx=∫u10(13du)=13∫u10du.This evaluates as13∫u10du=13◻+c=◻+C

Answer

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Now, if $u = x^{3} + 6$ , then $\int x^{2} {(x^{3} + 6)}^{10} d x = \int u^{10} (\frac{1}{3} d u) = \frac{1}{3} \int u^{10} d u$ .
This evaluates as
$\frac{1}{3} \int u^{10} d u = \frac{1}{3} ◻ + c = ◻ + C$