SATHEE: Indefinite Integration 1 Question 50

Indefinite Integration 1 Question 50

51. The option(s) with the values of $a$ and $L$ that satisfy the equation $\frac{\int_{0}^{4 π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t}{\int_{0}^{π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t} = L$ , is/are

(2015 Adv.)

(a) $a = 2, L = \frac{e^{4 π} - 1}{e^{π} - 1}$

(b) $a = 2, L = \frac{e^{4 π} + 1}{e^{π} + 1}$

(d) $a = 4, L = \frac{e^{4 π} + 1}{e^{π} + 1}$

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Solution:

Let $I_{1} = \int_{0}^{4 π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t$

$= \int_{0}^{π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t$

$\begin{aligned} + \int_{π}^{2 π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t \\ + \int_{2 π}^{3 π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t \\ + \int_{3 π}^{4 π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t \end{aligned}$

$∴ I_{1} = I_{2} + I_{3} + I_{4} + I_{5}$

Now, $I_{3} = \int_{π}^{2 π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t$

Put $t = π + x \Rightarrow d t = d x$

$∴ I_{3} = \int_{0}^{π} e^{π + x} \cdot (\sin^{6} a t + \cos^{4} a t) d t = e^{π} \cdot I_{2} \dots$ (ii)

Now, $I_{4} = \int_{2 π}^{3 π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t$

Put $t = 2 π + x \Rightarrow d t = d x$

$∴ I_{4} = \int_{0}^{π} e^{x + 2 π} (\sin^{6} a t + \cos^{4} a t) d t = e^{2 π} \cdot I_{2}$

and $I_{5} = \int_{3 π}^{4 π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t$

Put $t = 3 π + x$

$∴ I_{5} = \int_{0}^{π} e^{3 π + x} (\sin^{6} a t + \cos^{4} a t) d t = e^{3 π} \cdot I_{2}$ From Eqs. (i), (ii), (iii) and (iv), we get

$\begin{aligned} I_{1} = I_{2} + e^{π} \cdot I_{2} + e^{2 π} \cdot I_{2} + e^{3 π} \cdot I_{2} = (1 + e^{π} + e^{2 π} + e^{3 π}) I_{2} \\ ∴ L = \frac{\int_{0}^{4 π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t}{\int_{0}^{π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t} \\ = (1 + e^{π} + e^{2 π} + e^{3 π}) \\ = \frac{1 \cdot (e^{4 π} - 1)}{e^{π} - 1} for a \in R \end{aligned}$

Indefinite Integration 1 Question 50

51. The option(s) with the values of $a$ and $L$ that satisfy the equation $\frac{\int_{0}^{4 π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t}{\int_{0}^{π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t} = L$ , is/are

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Indefinite Integration 1 Question 50

51. The option(s) with the values of a and L that satisfy the equation ∫04πet(sin6⁡at+cos4⁡at)dt∫0πet(sin6⁡at+cos4⁡at)dt=L, is/are

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51. The option(s) with the values of $a$ and $L$ that satisfy the equation $\frac{\int_{0}^{4 π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t}{\int_{0}^{π} e^{t} (\sin^{6} a t + \cos^{4} a t) d t} = L$ , is/are