Question

Integrate x/x^2+1dx 2to1

Answer 1

EXPLANATION.

$\sf \implies \displaystyle\int\limits^2_1 {\dfrac{x}{x^{2} + 1} } \, dx$

As we know that,

By using substitution method,

Let we assume that,

⇒ (x² + 1) = t.

⇒ 2x dx = dt.

⇒ x dx = dt/2.

Put the values in the equation, we get.

$\sf \implies \displaystyle\int\limits^2_1 {\dfrac{dt}{2t} } \,$

$\sf \implies \dfrac{1}{2} \displaystyle\int\limits^2_1 {\dfrac{dt}{t} } \,$

$\sf \implies \dfrac{1}{2} \displaystyle \ \bigg[log |t| \bigg]_{1}^{2}$

Put the values of t = x² + 1 in equation, we get.

$\sf \implies \dfrac{1}{2} \displaystyle \bigg[ log | x^{2} + 1 | \bigg]_{1}^{2}$

As we know that,

First we put upper limit then we put lower limit in definite integration, we get.

$\sf \implies \dfrac{1}{2} \bigg[ log | (2)^{2} + 1 | \bigg] \ - \ \dfrac{1}{2} \bigg[ log | (1)^{2} + 1 | \bigg]$

$\sf \implies \dfrac{1}{2} \bigg[ log (4 + 1) \bigg] - \dfrac{1}{2} \bigg[ log(1 + 1) \bigg]$

$\sf \implies \dfrac{1}{2} \bigg[ log (5) \bigg] - \dfrac{1}{2} \bigg[ log (2) \bigg].$

$\sf \implies \dfrac{1}{2} \bigg[ log (5) - log(2) \bigg].$

                                                                                                                         

MORE INFORMATION.

Some important formula.

$\sf \implies (1)= \displaystyle \int\limits^\frac{\pi}{2} _0 {log (sinx)} \, dx = \int\limits^\frac{\pi}{2} _0 {log(cosx)} \, dx = - \bigg(\dfrac{\pi}{2} \bigg)log(2).$

$\sf \implies(2) = \displaystyle \int\limits^\frac{\pi}{2} _0 {sin^{n} x} \, dx = \int\limits^\frac{\pi}{2} _0 {cos^{n}x } \, dx = \dfrac{(n - 1)}{n} .\dfrac{(n - 3)}{(n - 2)} ,,,\frac{2}{3} . 1 \ (n \ is \ odd)$

$\sf \implies(3) = \displaystyle \int\limits^\frac{\pi}{2} _0 {sin^{n} x} \, dx = \int\limits^\frac{\pi}{2} _0 {cos^{n}x } \, dx = \dfrac{(n - 1)}{n} . \frac{(n - 3)}{(n - 2)} ,,,,, \dfrac{1}{2} \times \frac{\pi}{2} \ (n \ is \ even).$

$\sf \displaystyle \int\limits^\frac{\pi}{2} _0 {sin^{m} x \ cos^{n}x } \, dx = \dfrac{(m - 1)(m - 3),,(2 \ or \ 1)(n - 1)(n - 3),,(2 \ or 1 )_}{(m + n)(m + n - 2),,(2 \ or 1)} \times ( 1 \ or \pi/2).$

It is important to note that we multiply by (π/2) when both m and n are even.