The value of integral ∫x⁹/(4x²+1)⁶ dx=___?

 

In order to find value of integral \[\int \frac{x^{9}}{\left ( 4x^{2}+1 \right )^{6}} dx\]

we are going to use substitution method .

\[\mathbf{Put\; \; \; } 4x^{2}+1=u.\\ \mathbf{Differentiating \; w.r.t.x,\; \; } 2.4x=du\Rightarrow x.dx=\frac{du}{8}.\\ \mathbf{And\; \; } 4 x^{2}=u-1\Rightarrow x^{2}=\frac{u-1}{4}\]

Now we have substitute these values before that consider the integral,

\[\begin{align*} \int \frac{x^{9}}{\left ( 4x^{2}+1 \right )^{6}} dx &= \int \frac{(x^{2})^{4}.x}{(4x^{2}+1)^{6}} \; dx \\ &= \int \frac{(\frac{u-1}{4})^{4}}{u^{6}} \frac{du}{8} \\ &= \int \frac{({u-1})^{4}}{4^{4}u^{4}u^{2}} \frac{du}{8}\\ &= \frac{1}{4^{4}.8} \int \left [ \frac{u-1}{u} \right ]^{4}\frac{1}{u^{2}} \; du\\ &=\frac{1}{4^{5}.2} \int \left [1 -\frac{1}{u} \right ]^{4}\frac{1}{u^{2}} \; du \end{align*}\]

Again by substituting 1-1/u=t.

\[\mathbf{Differentiating\; w.r.t\; u:} \; \; -(-1 u^{-2})=\frac{1}{u^{2}}=\frac{dt}{du} \Rightarrow \frac{1}{u^{2}} du=dt\]

Hence

\[\begin{align*} \frac{1}{4^{5}.2} \int \left [1 -\frac{1}{u} \right ]^{4}\frac{1}{u^{2}} \; du &= \frac{1}{4^{5}.2} \int \left [t \right ]^{4} \; dt\\ &= \frac{1}{4^{5}.2} \left [\frac{t^{5}}{5} +C\right ] \; \\ &= \frac{t^{5}}{4^{5}.10} +C \end{align*}\]

Now for finding the value of integral we have to substitute values of t and u.

 \[\begin{align*} \therefore \int \frac{x^{9}}{\left ( 4x^{2}+1 \right )^{6}}\; dx&= \frac{t^{5}}{10.4^{5}} +C\\ &= \frac{[1-\frac{1}{u}]^{5}}{10.4^{5}} +C\; \; (\mathbf{by \; substuiting \; t=1-\frac{1}{u}})\\ &= \frac{1}{10.4^{5}}\left [ 1-\frac{1}{4x^{2}+1} \right ]^{5} +C\; \; (\mathbf{by \; substuiting \; u=\frac{1}{4x^{2}+1}})\\ &=\frac{1}{10.4^{5}}\left [ \frac{4x^{2}+1-1}{4x^{2}+1} \right ]^{5} +C\\ &= \frac{1}{10.4^{5}}\left [ \frac{4x^{2}}{4x^{2}+1} \right ]^{5} +C\\ &= \frac{1}{10.4^{5}}\left [ \frac{4}{4+\frac{1}{x^{2}}} \right ]^{5} +C\; (Dividing \; by\; x^{2})\\ &= \frac{4^{5}}{10.4^{5}}\left [ \frac{1}{4+\frac{1}{x^{2}}} \right ]^{5} +C\;(Cancelling \; \; 4^{5}) \\&= \frac{1}{10.}\left [ {4+\frac{1}{x^{2}}} \right ]^{-5} +C\; \end{align*}\]

\[\begin{align*} \therefore \int \frac{x^{9}}{\left ( 4x^{2}+1 \right )^{6}}\; dx &= \frac{1}{10.}\left [ {4+\frac{1}{x^{2}}} \right ]^{-5} +C\; \end{align*}\]

Hence option (D) is correct

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