Integrate the following functions- -cos x-1-sin x d x

∫cos x/√(1+sin x)dx=∫ (1+sin x)^ 1/2cos x dx Let 1+sin x =t ⇒ cos x =dt/dx⇒ dx =dt/cos x ∴∫ (1+sin x)^ 1/2cos x dx =∫ (t)^ 1/2cos x dt/dx=t^ 1/2+1/( 1/2+1)+C = ...

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Standard XII

Mathematics

Integration of Irrational Algebraic Fractions - 2

Integrate the...

Question

Integrate the following functions.
$\int \frac{c o s x}{\sqrt{1 + s i n x}} d x .$

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Solution

$\int \frac{c o s x}{\sqrt{1 + s i n x}} d x = \int (1 + s i n x)^{- \frac{1}{2}} c o s x d x$
Let $1 + s i n x = t \Rightarrow c o s x = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{c o s x}$
$∴ \int (1 + s i n x)^{- \frac{1}{2}} c o s x d x = \int (t)^{- \frac{1}{2}} c o s x \frac{d t}{d x} = \frac{t^{- \frac{1}{2} + 1}}{(- \frac{1}{2} + 1)} + C = 2 t^{\frac{1}{2}} + C = 2 \sqrt{1 + s i n x} + C$

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Similar questions

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$\int \frac{c o s x}{(1 - s i n x) (2 - s i n x)} d x .$

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and

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Which one of the following is correct?

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Integrate the following functions. ∫cosx√1+sinxdx.

∫cosx√1+sinxdx=∫(1+sinx)−12cosxdx Let 1+sinx=t⇒cosx=dtdx⇒dx=dtcosx ∴∫(1+sinx)−12cosxdx=∫(t)−12cosxdtdx=t−12+1(−12+1)+C=2t12+C=2√1+sinx+C

Integrate the following functions.
$\int \frac{c o s x}{\sqrt{1 + s i n x}} d x .$