Question

Integration of ( $\sqrt{tanx} + \sqrt{cotx}$ )

Answer 1

Answer:

$\displaystyle\mathrm{\int (\sqrt{tanx}+\sqrt{cotx})dx=2\sqrt{tanx}+C}$

Solution:

Now, $\displaystyle\mathrm{\int (\sqrt{tanx}+\sqrt{cotx})dx}$

$\displaystyle\mathrm{=\int \left(\sqrt{tanx}+\frac{1}{\sqrt{tanx}}\right)dx}$

$\displaystyle\mathrm{=\int \frac{tanx+1}{\sqrt{tanx}}dx}$

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We consider, tanx = z²

⇒ sec²x dx = 2z dz

⇒ (1 + tan²x) dx = 2z dz

⇒ (1 + z²) dx = 2z dz

⇒ dx = 2z dz /(1 + z²)

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$\displaystyle\mathrm{=\int \frac{z^{2}+1}{z}\:\frac{2z}{1+z^{2}}\:dz}$

$\displaystyle\mathrm{=2\int dz}$

= 2z + C , where C is integral constant

$\displaystyle\mathrm{=2\sqrt{tanx}+C}$

$\displaystyle\to \mathrm{\int (\sqrt{tanx}+\sqrt{cotx})dx=2\sqrt{tanx}+C}$

Answer 2

Answer:

$\huge \bold \red{answer}$

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