Question

Integrate the above attachment
#Class 12th
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Answer 1

Your answer is in the attachment :)

Answer 2

EXPLANATION.

$\sf \implies \displaystyle \int \dfrac{dx}{x^{2} + 6x + 18}$

As we know that,

If the denominator of quadratic equation is not factorizes then change it into perfect square, we get.

$\sf \implies \displaystyle \int \dfrac{dx}{(x + 3)^{2} + 9}$

$\sf \implies \displaystyle \int \dfrac{dx}{(x + 3)^{2} + (3)^{2} }$

As we know that,

Formula of :

$\sf \implies \displaystyle \int \dfrac{dx}{x^{2} + a^{2} } \ = \dfrac{1}{a} tan^{-1} \bigg(\dfrac{x}{a} \bigg) + C$

Using this formula in equation, we get.

$\sf \implies \displaystyle \dfrac{1}{3} tan^{-1} \bigg(\dfrac{x + 3}{3} \bigg) + C$

$\sf \implies \displaystyle \int \dfrac{dx}{x^{2} + 6x + 18} \ = \dfrac{1}{3} tan^{-1} \bigg(\dfrac{x + 3}{3} \bigg) + C.$

                                                                                                                     

MORE INFORMATION.

$\sf (1) = \displaystyle \int \dfrac{p (sin x) + q (cos x)}{a (sin x) + b (cos x)} dx$

$\sf (2) = \displaystyle \int \dfrac{p (sin x)}{a (sin x) + b (cos x)} dx$

$\sf (3) = \displaystyle \int \dfrac{q (cos x)}{a (sin x) + b (cos x)} dx$

For their integration, we first express Nr. as follows.

Nr = A (Dr) + B (derivative of Dr).

Then integral = Ax + B ㏒(Dr) + C.