Calculus-Integration

Question

$\int x^7.e^{x^4}dx$

How to do this?

Answer 1

Let $I=\int x^{7}.e^{x^{4}}dx$

let $x^{4}=t$

differentiate both side with respect to $'t'$

    $4x^{3}dx=dt$

   $x^{3}dx=\frac{dt}{4}$

Now, $I=\int x^{4}.e^{x^{4}}.x^{3}dx$

        $I=\frac{1}{4} \int t.e^{t}dt$

         $I=\frac{1}{4}[t.\int e^{t}dt-\int [ \frac{\mathrm{d} }{\mathrm{d} t}(t)(\int e^{t}dt)]dt]$

         $I=\frac{1}{4}[t.e^{t}-[\int1.e^{t}dt]]$

         $I=\frac{1}{4}[t.e^{t}-[\int e^{t}dt]]$

         $I=\frac{1}{4}[t.e^{t}-e^{t}]+C$

         $I=\frac{1}{4}e^{t}(t-1)+C$

       Put $t=x^{4}$

     $I=\frac{1}{4}e^{x^{4}}(x^{4}-1)+C$

                                                          $(OR)$

 

Comments

You have missed divided by 4 in the subsequent steps after I=1/4∫t.e^tdt. Otherwise it is matching with mine.

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oh, sorry now i edit

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Nice approch...