Integral of $$$9 \cdot 15^{- x} x^{2}$$$

Find $$$\int 9 \cdot 15^{- x} x^{2}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=9$$$ and $$$f{\left(x \right)} = 15^{- x} x^{2}$$$:

$${\color{red}{\int{9 \cdot 15^{- x} x^{2} d x}}} = {\color{red}{\left(9 \int{15^{- x} x^{2} d x}\right)}}$$

For the integral $$$\int{15^{- x} x^{2} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=x^{2}$$$ and $$$\operatorname{dv}=15^{- x} dx$$$.

Then $$$\operatorname{du}=\left(x^{2}\right)^{\prime }dx=2 x dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{15^{- x} d x}=- \frac{15^{- x}}{\ln{\left(15 \right)}}$$$ (steps can be seen »).

Therefore,

$$9 {\color{red}{\int{15^{- x} x^{2} d x}}}=9 {\color{red}{\left(x^{2} \cdot \left(- \frac{15^{- x}}{\ln{\left(15 \right)}}\right)-\int{\left(- \frac{15^{- x}}{\ln{\left(15 \right)}}\right) \cdot 2 x d x}\right)}}=9 {\color{red}{\left(- \int{\left(- \frac{2 \cdot 15^{- x} x}{\ln{\left(15 \right)}}\right)d x} - \frac{15^{- x} x^{2}}{\ln{\left(15 \right)}}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=- \frac{2}{\ln{\left(15 \right)}}$$$ and $$$f{\left(x \right)} = 15^{- x} x$$$:

$$- 9 {\color{red}{\int{\left(- \frac{2 \cdot 15^{- x} x}{\ln{\left(15 \right)}}\right)d x}}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}} = - 9 {\color{red}{\left(- \frac{2 \int{15^{- x} x d x}}{\ln{\left(15 \right)}}\right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}}$$

For the integral $$$\int{15^{- x} x d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=x$$$ and $$$\operatorname{dv}=15^{- x} dx$$$.

Then $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{15^{- x} d x}=- \frac{15^{- x}}{\ln{\left(15 \right)}}$$$ (steps can be seen »).

The integral can be rewritten as

$$\frac{18 {\color{red}{\int{15^{- x} x d x}}}}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}}=\frac{18 {\color{red}{\left(x \cdot \left(- \frac{15^{- x}}{\ln{\left(15 \right)}}\right)-\int{\left(- \frac{15^{- x}}{\ln{\left(15 \right)}}\right) \cdot 1 d x}\right)}}}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}}=\frac{18 {\color{red}{\left(- \int{\left(- \frac{15^{- x}}{\ln{\left(15 \right)}}\right)d x} - \frac{15^{- x} x}{\ln{\left(15 \right)}}\right)}}}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=- \frac{1}{\ln{\left(15 \right)}}$$$ and $$$f{\left(x \right)} = 15^{- x}$$$:

$$\frac{18 \left(- {\color{red}{\int{\left(- \frac{15^{- x}}{\ln{\left(15 \right)}}\right)d x}}} - \frac{15^{- x} x}{\ln{\left(15 \right)}}\right)}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}} = \frac{18 \left(- {\color{red}{\left(- \frac{\int{15^{- x} d x}}{\ln{\left(15 \right)}}\right)}} - \frac{15^{- x} x}{\ln{\left(15 \right)}}\right)}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}}$$

Let $$$u=- x$$$.

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

The integral becomes

$$\frac{18 \left(\frac{{\color{red}{\int{15^{- x} d x}}}}{\ln{\left(15 \right)}} - \frac{15^{- x} x}{\ln{\left(15 \right)}}\right)}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}} = \frac{18 \left(\frac{{\color{red}{\int{\left(- 15^{u}\right)d u}}}}{\ln{\left(15 \right)}} - \frac{15^{- x} x}{\ln{\left(15 \right)}}\right)}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = 15^{u}$$$:

$$\frac{18 \left(\frac{{\color{red}{\int{\left(- 15^{u}\right)d u}}}}{\ln{\left(15 \right)}} - \frac{15^{- x} x}{\ln{\left(15 \right)}}\right)}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}} = \frac{18 \left(\frac{{\color{red}{\left(- \int{15^{u} d u}\right)}}}{\ln{\left(15 \right)}} - \frac{15^{- x} x}{\ln{\left(15 \right)}}\right)}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}}$$

Apply the exponential rule $$$\int{a^{u} d u} = \frac{a^{u}}{\ln{\left(a \right)}}$$$ with $$$a=15$$$:

$$\frac{18 \left(- \frac{{\color{red}{\int{15^{u} d u}}}}{\ln{\left(15 \right)}} - \frac{15^{- x} x}{\ln{\left(15 \right)}}\right)}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}} = \frac{18 \left(- \frac{{\color{red}{\frac{15^{u}}{\ln{\left(15 \right)}}}}}{\ln{\left(15 \right)}} - \frac{15^{- x} x}{\ln{\left(15 \right)}}\right)}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}}$$

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

$$\frac{18 \left(- \frac{15^{{\color{red}{u}}}}{\ln{\left(15 \right)}^{2}} - \frac{15^{- x} x}{\ln{\left(15 \right)}}\right)}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}} = \frac{18 \left(- \frac{15^{{\color{red}{\left(- x\right)}}}}{\ln{\left(15 \right)}^{2}} - \frac{15^{- x} x}{\ln{\left(15 \right)}}\right)}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}}$$

Therefore,

$$\int{9 \cdot 15^{- x} x^{2} d x} = \frac{18 \left(- \frac{15^{- x} x}{\ln{\left(15 \right)}} - \frac{15^{- x}}{\ln{\left(15 \right)}^{2}}\right)}{\ln{\left(15 \right)}} - \frac{9 \cdot 15^{- x} x^{2}}{\ln{\left(15 \right)}}$$

Simplify:

$$\int{9 \cdot 15^{- x} x^{2} d x} = - \frac{9 \cdot 225^{x} 3375^{- x} \left(x^{2} \ln{\left(15 \right)}^{2} + 2 x \ln{\left(15 \right)} + 2\right)}{\ln{\left(15 \right)}^{3}}$$

Add the constant of integration:

$$\int{9 \cdot 15^{- x} x^{2} d x} = - \frac{9 \cdot 225^{x} 3375^{- x} \left(x^{2} \ln{\left(15 \right)}^{2} + 2 x \ln{\left(15 \right)} + 2\right)}{\ln{\left(15 \right)}^{3}}+C$$

$$$\int 9 \cdot 15^{- x} x^{2}\, dx = - \frac{9 \cdot 225^{x} 3375^{- x} \left(x^{2} \ln^{2}\left(15\right) + 2 x \ln\left(15\right) + 2\right)}{\ln^{3}\left(15\right)} + C$$$A