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F(x)=5^(x)

f(x)=F^(')(x)

int_(0)^(3)f(x)dx=

$F(x)=5^{x}$ $\newline$ $f(x)=F^{\prime}(x)$ $\newline$ $\int_{0}^{3} f(x) d x=$

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Evaluate definite integrals using the power rule

Full solution

Q.

F(x)=5^{x}

\newline

f(x)=F^{\prime}(x)

\newline

\int_{0}^{3} f(x) d x=

Integrate from $0$ to $3$ : Now, we integrate $f(x)$ from $0$ to $3$ . $\newline$ $\int_{0}^{3} \ln(5) \cdot 5^{x} \, dx$ $\newline$ Let's pull out the constant $\ln(5)$ . $\newline$ $\ln(5) \cdot \int_{0}^{3} 5^{x} \, dx$
Pull out constant $\ln(5)$ : To integrate $5^x$ , we use the formula $\int a^x \, dx = \frac{a^x}{\ln(a)} + C$ . $\int 5^x \, dx = \frac{5^x}{\ln(5)} + C$ Now we apply the limits from $0$ to $3$ . $\ln(5) \times \left[\frac{5^3}{\ln(5)} - \frac{5^0}{\ln(5)}\right]$
Apply integration formula: Simplify the expression. $\newline$ $\ln(5) \times \left[\frac{125}{\ln(5)} - \frac{1}{\ln(5)}\right]$ $\newline$ This simplifies to $125 - 1 = 124$ .

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