A calculus problem by Preetam Kandula

Calculus Level 4

Which of the following is an indefinite integral of ∫ 1 + 2 cot ⁡ x ( csc ⁡ x + cot ⁡ x ) d x \displaystyle \int \sqrt{1 + 2\cot x (\csc x + \cot x) } \, dx ?


Clarification: C C denotes the arbitrary constant of integration .

4 ln ⁡ ( sin ⁡ x 2 ) + C 4 \ln \left( \sin \frac x2\right) + C 2 ln ⁡ ( sin ⁡ x 2 ) + C 2 \ln \left( \sin \frac x2\right) + C 4 ln ⁡ ( cos ⁡ x 2 ) + C 4 \ln \left( \cos \frac x2\right) + C 2 ln ⁡ ( cos ⁡ x 2 ) + C 2 \ln \left( \cos \frac x2\right) + C

Preetam Kandula by Preetam Kandula , 21, India

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1 solution

Preetam Kandula
Apr 8, 2017

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice solution. We could also note that

1 + cos ⁡ ( x ) sin ⁡ ( x ) = 2 cos ⁡ 2 ( x 2 ) 2 sin ⁡ ( x 2 ) cos ⁡ ( x 2 ) = cos ⁡ ( x 2 ) sin ⁡ ( x 2 ) \dfrac{1 + \cos(x)}{\sin(x)} = \dfrac{2\cos^{2}(\frac{x}{2})}{2\sin(\frac{x}{2})\cos(\frac{x}{2})} = \dfrac{\cos(\frac{x}{2})}{\sin(\frac{x}{2})}

and then use the substitution u = sin ⁡ ( x 2 ) ⟹ d u = 1 2 sin ⁡ ( x 2 ) d x u = \sin(\frac{x}{2}) \Longrightarrow du = \frac{1}{2}\sin(\frac{x}{2}) dx .

Brian Charlesworth - 4 years, 2 months ago

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