Question

solve the integration​

Answer 1

EXPLANATION.

$\sf \implies \displaystyle \int \dfrac{x + 1}{4 + 5x - x^{2} } dx$

As we know that,

we can write equation as,

$\sf \implies \displaystyle - \int \dfrac{x + 1}{x^{2} - 5x - 4} dx$

Factorizes the equation into middle term splits, we get.

⇒ x² - 5x - 4.

⇒ x² - 4x - x - 4.

⇒ x(x - 4) + 1(x - 4).

⇒ (x + 1)(x - 4).

$\sf \implies \displaystyle - \int \dfrac{(x + 1)}{(x + 1)(x - 4)} dx$

$\sf \implies \displaystyle -\int \dfrac{dx}{x - 4} \ = - ln|x - 4| + C$

$\sf \implies \displaystyle \int \dfrac{x + 1}{4 + 5x - x^{2} } dx \ = -ln|x - 4| + C$

                                                                                                                         

MORE INFORMATION.

Integration by parts.

(1) = If u and v are two functions of x then,

⇒ ∫(u v)dx = u.(∫v dx) -∫[(du/dx) . (∫v dx)]dx.

From the first letter,

I = Inverse trigonometric functions.

L = Logarithmic functions.

A = Algebraic functions.

T = Trigonometric functions.

E = Exponential functions.

We get a word = ILATE.

Therefore, first arrange the functions in the order according to letters of this word and then integrate by parts.

(2) = ∫eˣ[f(x) + f'(x)]dx = eˣ f(x) + c.

(3) = ∫[x f'(x) + f(x)]dx = x f(x) + c.