Let-d-dx-F-x-e-sin-x-x-x-gt-0-If-1-4-2-e-sin-x-2-x-dx-F-k-F-1-then-one-of-the-possible-values-of-k-is-

Question Number 65961 by mmkkmm000m last updated on 06/Aug/19

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

If \underset{1}{\int^{4}} \frac{2 e^{\sin x^{2}}}{x} d x = F (k) - F (1), then one

of the possible values of k is

Answered by mr W last updated on 06/Aug/19

\underset{1}{\int^{4}} \frac{2 e^{\sin x^{2}}}{x} d x

= \underset{1}{\int^{4}} \frac{2 x e^{\sin x^{2}}}{x^{2}} d x

= \underset{1}{\int^{4}} \frac{e^{\sin x^{2}}}{x^{2}} d (x^{2})

= \underset{1}{\int^{16}} \frac{e^{\sin t}}{t} d t w i t h t = x^{2}

= {[F (t)]}_{1}^{16}

= F (16) - F (1)

= F (k) - F (1)

\Rightarrow k = 16

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