Evaluate
\frac{24x^{11}}{11}+36x^{9}+\frac{1458x^{7}}{7}+\frac{2187x^{5}}{5}+С
Differentiate w.r.t. x
3x^{4}\left(2x^{2}+9\right)^{3}
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\int 3x^{4}\left(8\left(x^{2}\right)^{3}+108\left(x^{2}\right)^{2}+486x^{2}+729\right)\mathrm{d}x
Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(2x^{2}+9\right)^{3}.
\int 3x^{4}\left(8x^{6}+108\left(x^{2}\right)^{2}+486x^{2}+729\right)\mathrm{d}x
To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.
\int 3x^{4}\left(8x^{6}+108x^{4}+486x^{2}+729\right)\mathrm{d}x
To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.
\int 24x^{10}+324x^{8}+1458x^{6}+2187x^{4}\mathrm{d}x
Use the distributive property to multiply 3x^{4} by 8x^{6}+108x^{4}+486x^{2}+729.
\int 24x^{10}\mathrm{d}x+\int 324x^{8}\mathrm{d}x+\int 1458x^{6}\mathrm{d}x+\int 2187x^{4}\mathrm{d}x
Integrate the sum term by term.
24\int x^{10}\mathrm{d}x+324\int x^{8}\mathrm{d}x+1458\int x^{6}\mathrm{d}x+2187\int x^{4}\mathrm{d}x
Factor out the constant in each of the terms.
\frac{24x^{11}}{11}+324\int x^{8}\mathrm{d}x+1458\int x^{6}\mathrm{d}x+2187\int x^{4}\mathrm{d}x
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{10}\mathrm{d}x with \frac{x^{11}}{11}. Multiply 24 times \frac{x^{11}}{11}.
\frac{24x^{11}}{11}+36x^{9}+1458\int x^{6}\mathrm{d}x+2187\int x^{4}\mathrm{d}x
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{8}\mathrm{d}x with \frac{x^{9}}{9}. Multiply 324 times \frac{x^{9}}{9}.
\frac{24x^{11}}{11}+36x^{9}+\frac{1458x^{7}}{7}+2187\int x^{4}\mathrm{d}x
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{6}\mathrm{d}x with \frac{x^{7}}{7}. Multiply 1458 times \frac{x^{7}}{7}.
\frac{24x^{11}}{11}+36x^{9}+\frac{1458x^{7}}{7}+\frac{2187x^{5}}{5}
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}. Multiply 2187 times \frac{x^{5}}{5}.
\frac{24x^{11}}{11}+36x^{9}+\frac{1458x^{7}}{7}+\frac{2187x^{5}}{5}+С
If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.
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