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  • Evaluate the definite integrals in Exercises 1 to 20. Integral 2 to 3 dx/x^2 - 1

Evaluate the definite integrals in Exercises 1 to 20.

    Q11.    \int_2^3 \frac{dx}{x^2 -1 }

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Given integral: \int_2^3 \frac{dx}{x^2 -1 }

Consider the integral \int \frac{dx}{x^2 -1 }

\int \frac{dx}{x^2 -1 } = \frac{1}{2}\log\left | \frac{x-1}{x+1} \right |

So, we have the function of x, f(x) =\frac{1}{2}\log\left | \frac{x-1}{x+1} \right |

Now, by Second fundamental theorem of calculus, we have

I = f(3) -f(2)

= \frac{1}{2}\left \{ \log\left | \frac{3-1}{3+1} \right | - \log\left | \frac{2-1}{2+1} \right | \right \}

= \frac{1}{2}\left \{ \log\left | \frac{2}{4} \right | -\log\left | \frac{1}{3} \right | \right \}

= \frac{1}{2}\left \{ \log \frac{1}{2} -\log \frac{1}{3} \right \} = \frac{1}{2}\log\frac{3}{2}

Posted by

Divya Prakash Singh

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