Question

Integration of x-5 where lower limit is 2
and upper limit is 8​

Answer 1

Step-by-step explanation:

first,,, integrate (x-5) with respect to x and after put the limits.

Answer 2

EXPLANATION.

$\sf \implies \displaystyle\int\limits^8_2 {(x - 5)} \, dx$

As we know that,

We can write equation as,

$\sf \implies \displaystyle\int\limits^8_2 {(x)} \, dx \ -\ 5 \displaystyle\int\limits^8_2 dx$

As we know that,

Formula of :

⇒ ∫xⁿdx = xⁿ⁺¹/n + 1 + c, (n ≠ -1).

Using this formula in equation, we get.

$\sf \implies \displaystyle\int\limits^8_2 \bigg[\dfrac{x^{2} }{2} \ - 5x\bigg]$

In definite integration,

First put the upper limits then put the lower limit, we get.

$\sf \implies \bigg[\dfrac{(8)^{2} }{2} \ - \ 5(8)\bigg] \ - \ \bigg[\dfrac{(2)^{2} }{2} \ - \ 5(2)\bigg]$

$\sf \implies \bigg[\dfrac{64}{2} - 40\bigg] \ - \ \bigg[\dfrac{4}{2} \ - \ 10\bigg]$

$\sf \implies \bigg[32 \ - \ 40 \bigg] \ - \ \bigg[2 \ - \ 10\bigg]$

$\sf \implies \bigg[-8\bigg] \ - \ \bigg[-8\bigg].$

$\sf \implies - 8 + 8 = 0.$

$\sf \implies \displaystyle\int\limits^8_2 {(x - 5)} \, dx = 0.$

                                                                                                                   

MORE INFORMATION.

Some important formula.

$\sf \implies \displaystyle\int\limits^\frac{\pi}{2} _0 {log (sinx)} \, dx = \displaystyle\int\limits^\frac{\pi}{2} _0 log(cosx)dx = - \bigg(\dfrac{\pi}{2}\bigg)log2.$

WALLI'S FORMULA [define of m and n].

$\sf \implies \displaystyle\int\limits^\frac{\pi}{2} _0sin^{n} dx = \displaystyle\int\limits^\frac{\pi}{2} _0 cos^{n} dx = \dfrac{(n - 1)}{n} .\dfrac{(n - 3)}{(n - 2)} ,,,, \dfrac{2}{3} .1 = (n \ is \ odd).$

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

$\sf \implies \displaystyle\int\limits^\frac{\pi}{2} _0sin^{m}x . cos^{n} x .dx= \dfrac{(m - 1)(m - 3),,,,(2 \ or \ 1)(n - 1)(n - 3),,,(2 \ or \ 1)}{(m + n)(m + n - 2),,,(2 \ or \ 1)} \ \times \bigg(1 \ or \ \dfrac{\pi}{2} \bigg)$

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