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Provide Solution For R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise  Revision Exercise Question 17 Maths Textbook Solution.

Answers (1)

Answer:

2 \log \left|e^{x}+1\right|-x+c

Given:

\int \frac{e^{x}-1}{e^{x}+1} d x

Hint:

Use partial fraction method.

Solution:

\int \frac{e^{x}-1}{e^{x}+1} d x

   \operatorname{let} e^{\mathrm{x}}=u

e^{x} d x=d u                                                            (differentiate with respect to x)

     d x=\frac{d u}{u}

now,

I=\int \frac{u-1}{u+1} \times \frac{d u}{u} \quad(p u t u \& d u)

 use partial fraction

u-1=A(1+u)+B u

u-1=A+A u+B u

A=-1 \& B+A=1 \quad \therefore B=2 \quad (on comparing)

So,\int \frac{e^{x}-1}{e^{x}+1} d x

=-\int \frac{1}{u}+\frac{2}{u+1} d u

=-\log |\mathrm{u}|+2 \log |\mathrm{u}+1|+\mathrm{c}

=-\log \left|\mathrm{e}^{\mathrm{x}}\right|+2 \log \left|\mathrm{e}^{\mathrm{x}}+1\right|+\mathrm{c}

=-2 \log \left|\mathrm{e}^{x}+1\right|-x+c \ldots(\operatorname{loge}=1)

 

 

Posted by

infoexpert21

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