Question Statement

$\int 5^{x}dx$

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Background and Explanation

These problems involve integrating exponential functions with arbitrary bases. Q10 applies the basic exponential integral formula directly, while Q11 requires substitution to handle the composite exponent $7y$.

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Solution

Let $I=\int 5^{x}dx$.

This is a standard exponential integral of the form $\int a^{x}dx$ where $a=5$. Applying the formula:

$I=\frac{5^x}{\ln 5}+c$

(Recall: $\int a^{x}dx=\frac{a^x}{\ln a}+c$ for any constant $a>0,a\ne 1$)

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Key Formulas or Methods Used

• Exponential Integral: $\int a^{x}dx=\frac{a^x}{\ln a}+c$ (for $a>0,a\ne 1$)
• Substitution Method (u-substitution): For composite functions $f(g(x))$, substitute $u=g(x)$ and $du=g^{\prime }(x)dx$ to simplify the integral
• Differential Substitution: If $u=ky$, then $du=kdy$ or $dy=\frac{du}{k}$

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Summary of Steps

($\int 5^{x}dx$):

1. Identify as basic exponential integral with base $a=5$
2. Apply formula: $\frac{5^x}{\ln 5}+c$