I can't quite understand your process, not being offensive but it looks a bit messy. (I prefer LaTeX more XD)
Checking your answer:\(\int _2^{7.5}3^{-x}\cdot\frac{100}{5^2}\cdot9\cdot31,500,000\space dx\)
Simplify & Pull the constant out:
\(=1,134,000,000\int _2^{7.5}3^{-x}dx\)
Now solving: \(\int3^{-x}dx\)
Substitute \(u=-x,\space dx=-du\)
\(=-\int3^udu\)
Apply exponential rule:
\(=-\frac{3^u}{\ln \left(3\right)}\)
Unsubstitude \(u\):
\(-\frac{3^{-x}}{\ln \left(3\right)}\)
So: \(\int3^{-x}dx=-\frac{3^{-x}}{\ln \left(3\right)}\)
Therefore: \(\int _2^{7.5}3^{-x}dx=\left(-\frac{3^{-7.5}}{\ln \left(3\right)}\right)-\left(-\frac{3^{-2}}{\ln \left(3\right)}\right)\)
\(\approx0.1009\)
\(Ans=0.1009\cdot1,134,000,000\)
\(\approx114420000 (Joules)\)
(Wait a minute, why is my answer different from yours @_@)
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