Integral of $$$\frac{25}{4 \left(\frac{3}{2} - x\right)^{2}}$$$

Find $$$\int \frac{25}{4 \left(\frac{3}{2} - x\right)^{2}}\, dx$$$.

Simplify the integrand:

$${\color{red}{\int{\frac{25}{4 \left(\frac{3}{2} - x\right)^{2}} d x}}} = {\color{red}{\int{\frac{25}{4 x^{2} - 12 x + 9} d x}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=25$$$ and $$$f{\left(x \right)} = \frac{1}{4 x^{2} - 12 x + 9}$$$:

$${\color{red}{\int{\frac{25}{4 x^{2} - 12 x + 9} d x}}} = {\color{red}{\left(25 \int{\frac{1}{4 x^{2} - 12 x + 9} d x}\right)}}$$

Complete the square (steps can be seen »): $$$4 x^{2} - 12 x + 9 = \left(2 x - 3\right)^{2}$$$:

$$25 {\color{red}{\int{\frac{1}{4 x^{2} - 12 x + 9} d x}}} = 25 {\color{red}{\int{\frac{1}{\left(2 x - 3\right)^{2}} d x}}}$$

Let $$$u=2 x - 3$$$.

Then $$$du=\left(2 x - 3\right)^{\prime }dx = 2 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{2}$$$.

Therefore,

$$25 {\color{red}{\int{\frac{1}{\left(2 x - 3\right)^{2}} d x}}} = 25 {\color{red}{\int{\frac{1}{2 u^{2}} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(u \right)} = \frac{1}{u^{2}}$$$:

$$25 {\color{red}{\int{\frac{1}{2 u^{2}} d u}}} = 25 {\color{red}{\left(\frac{\int{\frac{1}{u^{2}} d u}}{2}\right)}}$$

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

$$\frac{25 {\color{red}{\int{\frac{1}{u^{2}} d u}}}}{2}=\frac{25 {\color{red}{\int{u^{-2} d u}}}}{2}=\frac{25 {\color{red}{\frac{u^{-2 + 1}}{-2 + 1}}}}{2}=\frac{25 {\color{red}{\left(- u^{-1}\right)}}}{2}=\frac{25 {\color{red}{\left(- \frac{1}{u}\right)}}}{2}$$

Recall that $$$u=2 x - 3$$$:

$$- \frac{25 {\color{red}{u}}^{-1}}{2} = - \frac{25 {\color{red}{\left(2 x - 3\right)}}^{-1}}{2}$$

Therefore,

$$\int{\frac{25}{4 \left(\frac{3}{2} - x\right)^{2}} d x} = - \frac{25}{2 \left(2 x - 3\right)}$$

Simplify:

$$\int{\frac{25}{4 \left(\frac{3}{2} - x\right)^{2}} d x} = - \frac{25}{4 x - 6}$$

Add the constant of integration:

$$\int{\frac{25}{4 \left(\frac{3}{2} - x\right)^{2}} d x} = - \frac{25}{4 x - 6}+C$$

$$$\int \frac{25}{4 \left(\frac{3}{2} - x\right)^{2}}\, dx = - \frac{25}{4 x - 6} + C$$$A