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Which expression is equal to \[ \int e^{2x} \cdot e^{5} \, dx \]? A - HSC - SSCE Mathematics Extension 2 - Question 2 - 2021 - Paper 1
Question 2
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Which expression is equal to \[ \int e^{2x} \cdot e^{5} \, dx \]? A. \( \frac{1}{7} e^{5} e^{2x} - \frac{5}{7} \int e^{4x} \, dx \) B. \( \frac{1}{7} e^{5} e^{2... show full transcript
Worked Solution & Example Answer:Which expression is equal to \[ \int e^{2x} \cdot e^{5} \, dx \]? A - HSC - SSCE Mathematics Extension 2 - Question 2 - 2021 - Paper 1
Step 1
Step 2
Step B: Solve the integral
Answer
The integral ( \int e^{(2x + 5)} , dx ) can be solved using substitution. Let ( u = 2x + 5 ), then ( du = 2 , dx ) or ( dx = \frac{1}{2} du ). This transforms our integral:
[ \int e^{u} \cdot \frac{1}{2} , du = \frac{1}{2} e^{u} + C = \frac{1}{2} e^{(2x + 5)} + C ]
Step 3
Step C: Compare with options
Answer
We want to express our answer in the form presented in the choices. We notice that the expressions can be transformed or multiplied to bring out a common factor of ( \frac{1}{7} ). Only the option that fits this transformation and our derived answer is A. Thus, the matching expression is ( \frac{1}{7} e^{5} e^{2x} - \frac{5}{7} \int e^{4x} , dx ).