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Question Number 44712 by peter frank last updated on 03/Oct/18

	Answered by tanmay.chaudhury50@gmail.com last updated on 03/Oct/18

	$1) \int \frac{x^{n - 1}}{x^{n} (1 + x^{n})} d x$ $t = x^{n} d t = {n x}^{n - 1} d x$ $\frac{1}{n} \int \frac{d t}{t (1 + t)}$ $\frac{1}{n} [\int \frac{(1 + t) - t}{t (1 + t)} d t$ $\frac{1}{n} [\int \frac{d t}{t} - \int \frac{d t}{1 + t}]$ $\frac{1}{n} [l n (\frac{t}{1 + t})] + c$ $\frac{1}{n} [l n (\frac{x^{n}}{1 + x^{n}})] + c$
	Commented by peter frank last updated on 03/Oct/18

	$thank you$
	Answered by tanmay.chaudhury50@gmail.com last updated on 03/Oct/18

	$3) x + α = t$ $\int \frac{s i n (t - 2 α)}{s i n t} d t$ $\int \frac{s i n t c o s 2 α - c o s t s i n 2 α}{s i n t} d t$ $c o s 2 α \int d t - s i n 2 α \int \frac{c o s t}{s i n t} d t$ $c o s 2 α (t) - s i n 2 α l n s i n t + c$ $(x + α) c o s 2 α - s i n 2 α l n s i n (x + α) + c$
	Commented by tanmay.chaudhury50@gmail.com last updated on 03/Oct/18

	$p l s s e e i t a n d c h e k t h e a n s w e r y o u p r o v i d e d$ $a l o n g w i t h q u e s t i o n . . .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 03/Oct/18

	$i t h i n k a t t a c h e d a n s w e r n o t c o r r e v c t . . .$
	Commented by peter frank last updated on 03/Oct/18

	$true sir very sorry$
	Answered by tanmay.chaudhury50@gmail.com last updated on 03/Oct/18
	$2)∫(((x^4 −x)^(1/4) )/x^5 )dx ∫(({x^4 (1−(1/x^3 ))}^(1/4) )/x^5 ) ∫(((1−(1/x^3 ))^(1/4) )/x^4 )dx t=1−(1/x^3 ) dt=0−(−3)x^(−4) dx dt=((3dx)/x^4 ) ∫t^(1/4) ×(dt/3) (1/3)×(t^((1/4)+1) /((1/4)+1))+c (4/(15))t^(5/4) +c (4/(15))×(1−(1/x^3 ))^(5/4) +c$
	$2) \int \frac{{(x^{4} - x)}^{\frac{1}{4}}}{x^{5}} d x$ $\int \frac{{x^{4} (1 - \frac{1}{x^{3}})}^{\frac{1}{4}}}{x^{5}}$ $\int \frac{{(1 - \frac{1}{x^{3}})}^{\frac{1}{4}}}{x^{4}} d x$ $t = 1 - \frac{1}{x^{3}} d t = 0 - (- 3) x^{- 4} d x$ $d t = \frac{3 d x}{x^{4}}$ $\int t^{\frac{1}{4}} \times \frac{d t}{3}$ $\frac{1}{3} \times \frac{t^{\frac{1}{4} + 1}}{\frac{1}{4} + 1} + c$ $\frac{4}{15} t^{\frac{5}{4}} + c$ $\frac{4}{15} \times {(1 - \frac{1}{x^{3}})}^{\frac{5}{4}} + c$
	Commented by peter frank last updated on 03/Oct/18

	$thank you$
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