Class 12 Integrals · Conceptual Answer1 Marksindefinite_integration

Class 12 Maths · Integrals · Conceptual Answer #36

Evaluate \(\int 10^{- x} \left(2^{x + 1} - 5^{x - 1}\right)\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(- \frac{2 \cdot 5^{- x}}{\log{\left(5 \right)}} + \frac{2^{- x}}{5 \log{\left(2 \right)}}\) + C

Simplify the integrand to 2·5^(−x)−(1/5)2^(−x), then integrate the two exponentials using ∫q^(−x)dx=−q^(−x)/ln q.
Therefore the required integral is \(- \frac{2 \cdot 5^{- x}}{\log{\left(5 \right)}} + \frac{2^{- x}}{5 \log{\left(2 \right)}}\) + C.

[0.5 marks] Simplify the integrand to 2·5^(−x)−(1/5)2^(−x), then integrate the two exponentials using ∫q^(−x)dx=−q^(−x)/ln q.
[0.5 marks] Therefore the required integral is \(- \frac{2 \cdot 5^{- x}}{\log{\left(5 \right)}} + \frac{2^{- x}}{5 \log{\left(2 \right)}}\) + C.

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