Taking $x^{4}$ common from denominator
$=\int \frac{1 d x}{x^{2}\left(x^{4}\right)^{\frac{3}{4}}\left(1+\frac{1}{x^{4}}\right)^{\frac{3}{4}}}$
$=\int \frac{d x}{x^{2}\left(x^{3}\right)\left(1+\frac{1}{x^{4}}\right)^{\frac{3}{4}}}$
$=\int \frac{d t}{x^{5}\left(1+\frac{1}{x^{4}}\right)^{\frac{3}{4}}}$
Let $t = 1 + \frac{4}{{{x^4}}}$
$\frac{{dt}}{{dx}} = - \frac{4}{{{x^5}}}$
$-\frac{d t}{4}=\frac{d x}{x^{5}}$
Substituting value of $x$ and $d x$
$ = \frac{{ - 1}}{4}\int {\frac{{dt}}{{{t^{\frac{3}{4}}}}}} $
${=\frac{-1}{4} \int t^{\frac{-3}{4}} d t} $
${=\frac{-1}{4}\left[\frac{t^{\frac{-3}{4}}+1}{\frac{-3}{4}+1}\right]+C} $
${=\frac{-1}{4}\left(\frac{t^{\frac{1}{4}}}{\frac{1}{4}}\right)+C}$
$=-t^{\frac{1}{4}}+c$
$=-\left(1+\frac{1}{x^{4}}\right)^{\frac{1}{4}}+c$
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$l_1: \overrightarrow{ r }=(\hat{ i }-11 \hat{ j }-7 \hat{ k })+\lambda(\hat{ i }+2 \hat{ j }+3 \hat{ k }), \lambda \in R$ તથા
$l_2: \overrightarrow{ r }=(-\hat{ i }+\hat{ k })+\mu(2 \hat{ i }+2 \hat{ j }+\hat{ k }), \mu \in R$ ને લંબ છે.
જો $l$ અને $l_1$ નું છેદબિંદુ $P$ હોય તથા $P$ માંથી $l_2$ પરના લંબનો લંબપાદ $Q(\alpha, \beta, \lambda)$ હોય,તો $9(\alpha+\beta+\lambda)=.......$