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Question Number 185951 by manxsol last updated on 30/Jan/23

$$\int x^{3}ln(x+4)dx$$

Answered by SEKRET last updated on 30/Jan/23

$$\int {\mathbf{x}}^3\cdot \mathbf{ln}(\mathbf{x}+4)\mathbf{dx}=$$$$=\frac{{\mathbf{x}}^4}{4}\cdot \mathbf{ln}(\mathbf{x}+4)-\int \frac{{\mathbf{x}}^4}{4}\cdot \frac{1}{\mathbf{x}+4}\mathbf{dx}$$$$=\frac{{\mathbf{x}}^4}{4}\cdot \mathbf{ln}(\mathbf{x}+4)-\frac{1}{4}\cdot \int ({\mathbf{x}}^3-4{\mathbf{x}}^2+16\mathbf{x}+\frac{256}{\mathbf{x}+4}-64)\mathbf{dx}$$$$=\frac{{\mathbf{x}}^4}{4}\cdot \mathbf{ln}(\mathbf{x}-4)-\frac{1}{4}\cdot (\frac{{\mathbf{x}}^4}{4}-\frac{4{\mathbf{x}}^3}{3}+8{\mathbf{x}}^2+256\cdot \mathbf{ln}(\mathbf{x}+4)-64\mathbf{x})+\mathbf{C}$$

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