Integrate the following functions- e 2 x- e -2 x - e 2 x- e -2 x dx

∫e^2x e^ 2x/e^2x+e^ 2xdx Let e^2x+e^ 2x=t ⇒ 2e^2x 2e^ 2x=dt/dx⇒ dx =dt/2(e^2x e^ 2x) ∴∫e^2x e^ 2x/e^2x+e^ 2xdx =∫e^2x e^ 2x/tdt/2[e^2x e^ 2x] =1/2∫1/tdt =1/2lo ...

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Standard XII

Mathematics

Integration by Partial Fractions

Integrate the...

Question

Integrate the following functions.
$\int \frac{e^{2 x} - e^{- 2 x}}{e^{2 x} + e^{- 2 x}} d x .$

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Solution

$\int \frac{e^{2 x} - e^{- 2 x}}{e^{2 x} + e^{- 2 x}} d x$
Let $e^{2 x} + e^{- 2 x} = t \Rightarrow 2 e^{2 x} - 2 e^{- 2 x} = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{2 (e^{2 x} - e^{- 2 x})}$
$∴ \int \frac{e^{2 x} - e^{- 2 x}}{e^{2 x} + e^{- 2 x}} d x = \int \frac{e^{2 x} - e^{- 2 x}}{t} \frac{d t}{2 [e^{2 x} - e^{- 2 x}]} = \frac{1}{2} \int \frac{1}{t} d t = \frac{1}{2} l o g | t | + C = \frac{1}{2} l o g | e^{2 x} + e^{- 2 x} | + C$

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Integrate the following functions. ∫e2x−e−2xe2x+e−2xdx.

∫e2x−e−2xe2x+e−2xdx Let e2x+e−2x=t⇒2e2x−2e−2x=dtdx⇒dx=dt2(e2x−e−2x) ∴∫e2x−e−2xe2x+e−2xdx=∫e2x−e−2xtdt2[e2x−e−2x]=12∫1tdt=12log|t|+C=12log|e2x+e−2x|+C

Integrate the following functions.
$\int \frac{e^{2 x} - e^{- 2 x}}{e^{2 x} + e^{- 2 x}} d x .$