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Answer:

When you multiply monomials, first multiply the coefficients and then multiply the variables by adding the exponents. Note that when you multiply monomials with same base, you can add their exponents. This is called the Product of Powers Property.

Explanation:

I hope it helps.

[tex] \boxed{\boldsymbol{\rm{\dfrac{2}{\sqrt{3}}\arctan\left(\dfrac{x}{\sqrt{3x + 3}}\right) + C}}} [/tex]

Evaluate the indefinite integral

[tex] \displaystyle \rm \int \dfrac{x + 2}{(x^2 + 3x + 3)\sqrt{x + 1}}\ dx [/tex]

We can rewrite the given integral into

[tex] \displaystyle \rm \int \dfrac{ x + 2 }{((x + 1)(x + 2) + 1) \sqrt{x + 1}}\ dx [/tex]

By substitution, let [tex] \rm u = \sqrt{x + 1}.[/tex]

[tex] \begin{aligned} \rm u^2 &= \rm x + 1 \implies 2u\ du = dx \\ \rm u^2 + 1 &= \rm x + 2 \end{aligned} [/tex]

The integral becomes

[tex] \displaystyle \rm \int \dfrac{u^2 + 1}{(u^2(u^2 + 1)+1)\cancel{u}}\cdot 2\cancel{u}\ du [/tex]

[tex] = \displaystyle \rm 2 \int \dfrac{u^2 + 1}{u^4 + u^2 + 1}\ du [/tex]

Now, divide both numerator and denominator by [tex] \rm u^2. [/tex]

[tex] = \displaystyle \rm 2 \int \dfrac{1 + \dfrac{1}{u^2}}{u^2 + 1 + \dfrac{1}{u^2}}\ du [/tex]

[tex] = \displaystyle \rm 2 \int \dfrac{1 + \dfrac{1}{u^2}}{\left(u - \dfrac{1}{u}\right)^2 + 3}\ du [/tex]

[tex] \footnotesize \because \begin{array}{l} \rm u^2 + 1 + \dfrac{1}{u^2} = \left(u - \dfrac{1}{u}\right)^2 + 3 \quad and; \\ \\ \rm \dfrac{d}{du}\left(u - \dfrac{1}{u}\right) = 1 + \dfrac{1}{u^2} \end{array} [/tex]

[tex] = \rm \dfrac{2}{\sqrt{3}}\arctan{\left(\dfrac{u - \dfrac{1}{u}}{\sqrt{3}}\right)} + C [/tex]

[tex] = \rm \dfrac{2}{\sqrt{3}}\arctan{\left(\dfrac{\sqrt{x + 1} - \dfrac{1}{\sqrt{x + 1}}}{\sqrt{3}}\right)} + C [/tex]

Simplifying, we get

[tex] = \rm \dfrac{2}{\sqrt{3}}\arctan{\left(\dfrac{x}{\sqrt{3x + 3}}\right)} + C [/tex]

Thus,

[tex] \footnotesize \boxed{\displaystyle \rm{\int \dfrac{x + 2}{(x^2 + 3x + 3)\sqrt{x + 1}}\ dx = \dfrac{2}{\sqrt{3}}\arctan{\left(\dfrac{x}{\sqrt{3x + 3}}\right)} + C}} [/tex]

Answer:

The term economic environment refers to all the external economic factors that influence buying habits of consumers and businesses and therefore affect the performance of a company.

Explanation:

For me, I think the meaning in this picture is that our ideas in our mind are slowly fading.