Integrate the following functions-sin x-1-cos x2 d x -

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

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Standard XII

Mathematics

Theorems on Integration

Integrate the...

Question

Integrate the following functions.
$\int \frac{s i n x}{(1 + c o s x)^{2}} d x .$

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Solution

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

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Integrate the following functions. ∫sinx(1+cosx)2dx.

Let I=∫sinx(1+cosx)2dx Let 1+cosx=t⇒−sinx=dtdx⇒dx=dt−sinx ∴I=∫sinx(1+cosx)2dx=∫sinxt2×dt−sinx=−∫1t2dt=−∫t−2dt=−t−2+1−2+1+C=1t+C=11+cosx+C

Integrate the following functions.
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