Discussion Forum

 The options (s) with the values of a and L that  satidfy the equation  $\frac{\int_{0}^{4\pi} e^{t}(\sin^{6}at+\cos^{4}at)dt}{\int_{0}^{\pi} e^{t}(\sin^{6}at+\cos^{4}at)dt} =L$   is\are

Answer:

Explanation:

Let    $I_{1}=\int_{0}^{4\pi} e^{t}(\sin^{6}at+\cos^{6}at)dt$

 =$\int_{0}^{\pi} e^{t}(\sin^{6}at+\cos^{6}at)dt$

+  $\int_{\pi}^{2\pi} e^{t}(\sin^{6}at+\cos^{6}at)dt$

+  $\int_{2\pi}^{3\pi} e^{t}(\sin^{6}at+\cos^{6}at)dt$

+$\int_{3\pi}^{4\pi} e^{t}(\sin^{6}at+\cos^{6}at)dt$

    $\therefore$      $ I_{1}=I_{2}+I_{3}+I_{4}+I_{5}$       ..........(i)

Now,    $I_{3}=\int_{\pi}^{2\pi} e^{t}(\sin^{6}at+\cos^{6}at)dt$

 Put     $t=\pi+t\Rightarrow dt=dt$

          $\therefore$      $I_{3}=\int_{0}^{\pi} e^{\pi+t}(\sin^{6}at+\cos^{6}at)dt$

=  $e^{t}.I_{2}$            .........(ii)

Now,

          I₄= $\int_{2\pi}^{3\pi} e^{t}(\sin^{6}at+\cos^{6}at)dt$

               Put     $t=2\pi+t\Rightarrow dt=dt$

                        $\therefore$      $I_{4}=\int_{0}^{\pi} e^{2\pi+t}(\sin^{6}at+\cos^{6}at)dt$

=  $e^{2 \pi}.I_{2}$       ..........(iii)

and I₅=   $\int_{3\pi}^{4\pi} e^{t}(\sin^{6}at+\cos^{6}at)dt$

Put            $t=3\pi+t\Rightarrow dt=dt$

                        $\therefore$      $I_{5}=\int_{0}^{\pi} e^{3\pi+t}(\sin^{6}at+\cos^{6}at)dt$

=  $e^{3 \pi}.I_{2}$ .........(iv)

 From Eqs. (i), (ii) ,(iii) and (iv)  , we get

              $I_{1}=I_{2}+e^{\pi}I_{2}+e^{2\pi}I_{2}+e^{3\pi}I_{2}$

     $=I_{2}(1++e^{\pi}+e^{2\pi}+e^{3\pi})$

 $L= \frac{\int_{0}^{4\pi} e^{t}(\sin^{6}at+\cos^{6}at)dt}{\int_{0}^{\pi} e^{t}(\sin^{6}at+\cos^{6}at)dt}$

=     $(1+e^{\pi}+e^{2\pi}+e^{3\pi})$

                = $\frac{1-(e^{4\pi}-1)}{e^{\pi}-1} $   for    $a \epsilon R$