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

Evaluate the definite integrals in Exercises 1 to 20.

    Q13.    \int_2^3\frac{xdx}{x^2+1}

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Given integral: \int_2^3\frac{xdx}{x^2+1}

Consider the integral \int \frac{xdx}{x^2+1}

\int \frac{xdx}{x^2+1} = \frac{1}{2}\int \frac{2x}{x^2+1}dx =\frac{1}{2}\log(1+x^2)

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

Now, by Second fundamental theorem of calculus, we have

I = f(3) -f(2)

= \frac{1}{2}\left \{ \log(1+(3)^2)-\log(1+(2)^2) \right \}

= \frac{1}{2}\left \{ \log(10)-\log(5) \right \} = \frac{1}{2}\log\left ( \frac{10}{5} \right ) = \frac{1}{2}\log2

 

Posted by

Divya Prakash Singh

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