If-x-5-x-7-x-6-dx-p-ln-x-q-x-q-1-C-find-the-value-of-p-and-q-

Question Number 118230 by bobhans last updated on 16/Oct/20

If \int \frac{{(\sqrt{x})}^{5}}{{(\sqrt{x})}^{7} + x^{6}} dx = p \ln (\frac{x^{q}}{x^{q} + 1}) + C

find the value of p and q .

Answered by bemath last updated on 16/Oct/20

I = \int \frac{x^{2} \sqrt{x}}{x^{3} \sqrt{x} + x^{6}} d x = \int \frac{\sqrt{x}}{x \sqrt{x} + x^{4}} d x

I = \int \frac{d x}{x + x^{3} \sqrt{x}} .

l e t \sqrt{x} = u \to d x = 2 \sqrt{x} d u

I = \int \frac{2 u d u}{u^{2} + u^{7}} = \int \frac{2 d u}{u + u^{6}}

I = \int \frac{2 d u}{u (1 + u^{5})} = \int \frac{2 (1 + u^{5} - u^{5})}{u (1 + u^{5})} d u

= 2 \int (\frac{1}{u} - \frac{u^{4}}{1 + u^{5}}) d u

= 2 (\ln u - \frac{1}{5} \int \frac{d (1 + u^{5})}{1 + u^{5}})

= 2 (\ln u - \frac{1}{5} \ln (1 + u^{5})) + c

= \frac{2}{5} (5 \ln u - \ln (1 + u^{5})) + c

= \frac{2}{5} \ln (\frac{u^{5}}{1 + u^{5}}) + c = p \ln (\frac{x^{q}}{1 + x^{q}}) + c

= \frac{2}{5} \ln (\frac{x^{\frac{5}{2}}}{1 + x^{\frac{5}{2}}}) + c = p \ln (\frac{x^{q}}{1 + x^{q}}) + c

w e g e t {\begin{cases} p = \frac{2}{5} \\ q = \frac{5}{2} \end{cases}

Commented by bobhans last updated on 16/Oct/20

c o r r e c t!

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