SATHEE: Indefinite Integration 1 Question 55

Indefinite Integration 1 Question 55

56. Let $\frac{d}{d x} F (x) = \frac{e^{\sin x}}{x}, x > 0$ .

If $\int_{1}^{4} \frac{2 e^{\sin x^{2}}}{x} d x = F (k) - F (1)$ , then one of the possible values of $k$ is …..

$(1997, 2 M)$

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Solution:

Given, $\int_{1}^{4} \frac{2 e^{\sin x^{2}}}{x} d x = F (k) - F (1)$

Put $x^{2} = t$

$\Rightarrow 2 x d x = d t$

$\Rightarrow \int_{1}^{16} 2 \frac{e^{\sin t}}{t} \cdot \frac{d t}{2} = F (k) - F (1)$

$\Rightarrow \int_{1}^{16} \frac{e^{\sin t}}{t} d t = F (k) - F (1)$

$\begin{array}{lc} \Rightarrow & [F (t)]_{1}^{16} = F (k) - F (1) \\ ∵ \frac{d}{d x} F (x) = \frac{e^{\sin x}}{x}, given \\ \Rightarrow & F (16) - F (1) = F (k) - F (1) \\ ∴ & k = 16 \end{array}$

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Indefinite Integration 1 Question 55

56. Let ddxF(x)=esin⁡xx,x>0.

Solution:

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56. Let $\frac{d}{d x} F (x) = \frac{e^{\sin x}}{x}, x > 0$ .