Integral of $$$- \frac{11 x^{10}}{4} + 4 x^{3} - 7 x^{2}$$$

Find $$$\int \left(- \frac{11 x^{10}}{4} + 4 x^{3} - 7 x^{2}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- \frac{11 x^{10}}{4} + 4 x^{3} - 7 x^{2}\right)d x}}} = {\color{red}{\left(- \int{7 x^{2} d x} + \int{4 x^{3} d x} - \int{\frac{11 x^{10}}{4} d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=7$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$\int{4 x^{3} d x} - \int{\frac{11 x^{10}}{4} d x} - {\color{red}{\int{7 x^{2} d x}}} = \int{4 x^{3} d x} - \int{\frac{11 x^{10}}{4} d x} - {\color{red}{\left(7 \int{x^{2} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$\int{4 x^{3} d x} - \int{\frac{11 x^{10}}{4} d x} - 7 {\color{red}{\int{x^{2} d x}}}=\int{4 x^{3} d x} - \int{\frac{11 x^{10}}{4} d x} - 7 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\int{4 x^{3} d x} - \int{\frac{11 x^{10}}{4} d x} - 7 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4$$$ and $$$f{\left(x \right)} = x^{3}$$$:

$$- \frac{7 x^{3}}{3} - \int{\frac{11 x^{10}}{4} d x} + {\color{red}{\int{4 x^{3} d x}}} = - \frac{7 x^{3}}{3} - \int{\frac{11 x^{10}}{4} d x} + {\color{red}{\left(4 \int{x^{3} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=3$$$:

$$- \frac{7 x^{3}}{3} - \int{\frac{11 x^{10}}{4} d x} + 4 {\color{red}{\int{x^{3} d x}}}=- \frac{7 x^{3}}{3} - \int{\frac{11 x^{10}}{4} d x} + 4 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \frac{7 x^{3}}{3} - \int{\frac{11 x^{10}}{4} d x} + 4 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{11}{4}$$$ and $$$f{\left(x \right)} = x^{10}$$$:

$$x^{4} - \frac{7 x^{3}}{3} - {\color{red}{\int{\frac{11 x^{10}}{4} d x}}} = x^{4} - \frac{7 x^{3}}{3} - {\color{red}{\left(\frac{11 \int{x^{10} d x}}{4}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=10$$$:

$$x^{4} - \frac{7 x^{3}}{3} - \frac{11 {\color{red}{\int{x^{10} d x}}}}{4}=x^{4} - \frac{7 x^{3}}{3} - \frac{11 {\color{red}{\frac{x^{1 + 10}}{1 + 10}}}}{4}=x^{4} - \frac{7 x^{3}}{3} - \frac{11 {\color{red}{\left(\frac{x^{11}}{11}\right)}}}{4}$$

Therefore,

$$\int{\left(- \frac{11 x^{10}}{4} + 4 x^{3} - 7 x^{2}\right)d x} = - \frac{x^{11}}{4} + x^{4} - \frac{7 x^{3}}{3}$$

Simplify:

$$\int{\left(- \frac{11 x^{10}}{4} + 4 x^{3} - 7 x^{2}\right)d x} = x^{3} \left(- \frac{x^{8}}{4} + x - \frac{7}{3}\right)$$

Add the constant of integration:

$$\int{\left(- \frac{11 x^{10}}{4} + 4 x^{3} - 7 x^{2}\right)d x} = x^{3} \left(- \frac{x^{8}}{4} + x - \frac{7}{3}\right)+C$$

$$$\int \left(- \frac{11 x^{10}}{4} + 4 x^{3} - 7 x^{2}\right)\, dx = x^{3} \left(- \frac{x^{8}}{4} + x - \frac{7}{3}\right) + C$$$A