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Nezavi [6.7K]
3 years ago
9

Leslie bought 6.74 kilograms of apples and 5.2 kilograms of oranges. How much fruit did she buy in all?

Mathematics
1 answer:
mote1985 [20]3 years ago
5 0

Answer:

11.94

Step-by-step explanation:

In this problem you just need to add 6.74 with 5.2.

6.74 + 5.2 = 11.94

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Andreyy89

Answer:

\begin{aligned}\int{\frac{1}{1 + e^{x}}\cdot dx}= x - \ln(1 + e^{x}) + C\end{aligned}.

Step-by-step explanation:

The first derivative of the denominator 1 + e^{x} is e^{x}. Rewrite the fraction to obtain that expression on the numerator.

\begin{aligned}\frac{1}{1 + e^{x}} &= \frac{1 + e^{x}}{1 + e^{x}} - \frac{e^{x}}{1+e^{x}}\\&=1-\frac{e^{x}}{1+e^{x}}\end{aligned}.

In other words,

\begin{aligned}\int{\frac{1}{1 + e^{x}}\cdot dx} &= \int{dx} - \int{\frac{e^{x}}{1+e^{x}}\cdot dx}\end{aligned}.

Apply u-substitution on the integral \displaystyle \int{\frac{e^{x}}{1+e^{x}}\cdot dx}:

Let u = 1 + e^{x}. u > 1.

du = e^{x}\cdot dx.

\displaystyle \int{\frac{e^{x}}{1+e^{x}}\cdot dx} = \int{\frac{du}{u}} = \ln{|u|} = \ln{u} +C = \ln{(1+e^{x})}+C.

Therefore

\begin{aligned}\int{\frac{1}{1 + e^{x}}\cdot dx} &= \int{dx} - \int{\frac{e^{x}}{1+e^{x}}\cdot dx}\\ & = x - \ln{(1 + e^{x})}+C\end{aligned}.

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3 years ago
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