First slide

Methods of integration

Question

$Evaluate \int \sqrt{x^{2} + 2 x + 5} d x$

Moderate

A

$\frac{1}{2} (x + 1) \sqrt{x^{2} + 2 x + 5} + 2 \sin h^{- 1} x + C$
B

$\frac{1}{2} (x + 1) \sqrt{x^{2} + 2 x + 5} + 2 \cos h^{- 1} x + C$
C

$\frac{1}{2} (x + 1) \sqrt{x^{2} + 2 x + 5} - 2 \sin h^{- 1} x + C$
D

none of these

Solution

$\int \sqrt{x^{2} + 2 x + 5} d x = \int \sqrt{(x + 1)^{2} + 4} d x \int \sqrt{a^{2} + x^{2}} d x = \frac{x}{2} \sqrt{a^{2} + x^{2}} + \frac{a^{2}}{2} \sin h^{- 1} x + c \int \sqrt{(x + 1)^{2} + 4} d x = \frac{1}{2} (x + 1) \sqrt{(x + 1)^{2} + 2^{2}} + \frac{1}{2} \cdot (2)^{2} \sin h^{- 1} x + C = \frac{1}{2} (x + 1) \sqrt{x^{2} + 2 x + 5} + 2 \sin h^{- 1} x + C$

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