Question Statement

$\int 7^{7y}dy$

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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 $I=\int 7^{7y}dy$.

Since the exponent $7y$ is a linear function of $y$, we use substitution to simplify the integral.

Step 1: Substitute $7y=t$.

Step 2: Find the differential relationship. Taking differentials of both sides: $7dy=dt$ $dy=\frac{dt}{7}$

Step 3: Rewrite the integral in terms of $t$: $\begin{array}{rl}\int 7^{7y}dy& =\int 7^t\cdot \frac{dt}{7}\\ & =\frac{1}{7}\int 7^{t}dt\end{array}$

Step 4: Apply the exponential integral formula $\int a^{t}dt=\frac{a^t}{\ln a}$: $=\frac{1}{7}\cdot \frac{7^t}{\ln 7}+c$

Step 5: Substitute back $t=7y$: $=\frac{7^{7y}}{7\ln 7}+c$

This can also be written as $\frac{7^{7y-1}}{\ln 7}+c$.

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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

For Q10 ($\int 5^{x}dx$):

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

For Q11 ($\int 7^{7y}dy$):

1. Substitute $t=7y$ (inner function)
2. Calculate differential: $dy=\frac{dt}{7}$
3. Rewrite integral: $\frac{1}{7}\int 7^{t}dt$
4. Integrate: $\frac{1}{7}\cdot \frac{7^t}{\ln 7}+c$
5. Back-substitute $t=7y$: $\frac{7^{7y}}{7\ln 7}+c$