x-x-2-5-dx-3-x-x-2-5-dx-x-x-2-2-x-2-5-dx-

Question Number 72494 by aliesam last updated on 29/Oct/19

\frac{\int x \sqrt{x^{2} + 5} d x - 3 \int \frac{x}{\sqrt{x^{2} + 5}} d x}{\int \frac{x (x^{2} + 2)}{\sqrt{x^{2} + 5}} d x}

Commented by mathmax by abdo last updated on 29/Oct/19

l e t I = \int x \sqrt{x^{2} + 5} d x, J = \int \frac{x d x}{\sqrt{x^{2} + 5}}

K = \int \frac{x (x^{2} + 2)}{\sqrt{x^{2} + 5}} d x w e h a v e I - J = \int \frac{x (x^{2} + 5) - 3 x}{\sqrt{x^{2} + 5}} d x

= \int \frac{x^{3} + 5 x - 3 x}{\sqrt{x^{2} + 5}} d x = \int \frac{x (x^{2} + 2)}{\sqrt{x^{2} + 5}} d x \Rightarrow \frac{I - 3 J}{K} = 1

Answered by Tanmay chaudhury last updated on 29/Oct/19

I = \frac{I_{1} - 3 I_{2}}{I_{3}}

I_{1} = \int x \sqrt{x^{2} + 5} d x

= \frac{1}{2} \int {(x^{2} + 5)}^{\frac{1}{2}} \times d (x^{2} + 5)

= \frac{1}{2} \times \frac{{(x^{2} + 5)}^{\frac{3}{2}}}{\frac{3}{2}} + c_{1}

I_{2} = \int \frac{x}{\sqrt{x^{2} + 5}} d x

= \frac{1}{2} \int {(x^{2} + 5)}^{\frac{- 1}{2}} \times d (x^{2} + 5)

= \frac{1}{2} \times \frac{{(x^{2} + 5)}^{\frac{1}{2}}}{\frac{1}{2}} + c_{2}

\int \frac{x (x^{2} + 2)}{\sqrt{x^{2} + 5}} d x

k^{2} = x^{2} + 5 \to 2 k d k = 2 x d x

\int \frac{k^{2} - 5 + 2}{k} \times k d k

\int k^{2} - 3 d k

\frac{k^{3}}{3} - 3 k + c_{3}

\frac{{(x^{2} + 5)}^{\frac{3}{2}}}{3} - 3 {(x^{2} + 5)}^{\frac{1}{2}} + c_{3}

n o w p l s p u t t h e v a l u e s t o g e t I \dots

Commented by aliesam last updated on 29/Oct/19

t h a n k y o u s o m u c h s i r . g o d b l e s s y o u

Commented by Tanmay chaudhury last updated on 29/Oct/19

m o s t w e l c o m e s i r