Indefinite Integration
Standard Integrals
Premium Question
Grade 12
Question:
$\int \sqrt{x^2 + 2x + 5} \, dx$ equals:
(1) $\frac{x + 1}{2}\sqrt{x^2 + 2x + 5} + 2 \ln|x + 1 + \sqrt{x^2 + 2x + 5}| + C$
(2) $\frac{x + 1}{2}\sqrt{x^2 + 2x + 5} - 2 \ln|x + 1 + \sqrt{x^2 + 2x + 5}| + C$
(3) $(x + 1)\sqrt{x^2 + 2x + 5} + C$
(4) $\sqrt{x^2 + 2x + 5} + \ln|x + 1| + C$
Step-by-Step Solution
Key Concept: Complete the square inside the radical to write $x^2 + 2x + 5 = (x + 1)^2 + 2^2$, then apply the standard formula for $\int \sqrt{u^2 + a^2} \, du$.
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Correct Answer: (1)