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Let  \frac{d}{dx}F(x)=\left ( \frac{e^{sinx}}{x} \right ),x> 0.   If  \int_{1}^{4}\frac{3}{x}e^{sinx^{3}}\: dx=F(k)-F(1),   then one of the possible values of k is

  • Option 1)

    16

  • Option 2)

    63

  • Option 3)

    64

  • Option 4)

    15

 

Answers (1)

best_answer

As learnt in concept 

Integration by substitution -

The functions when on substitution of the variable of integration to some quantity gives any one of standard formulas.

 

 

- wherein

Since \int f(x)dx=\int f(t)dt=\int f(\theta )d\theta all variables must be converted into single variable ,\left ( t\, or\ \theta \right )

 

 

 

Indefinite integration -

It is inverse process of differentation.

\frac{d}{dx}\left \{ F(x) \right \}= f(x)

\therefore \int f(x)dx= F\left ( x \right )+C

 

- wherein

Where

\frac{d}{dx}F\left ( x \right ) is differential of F(x) w.r.t  x

 

 If we rewrite the statement, we get

F(x)=\int \frac{e^{sinx}}{x}dx

\int_{1}^{4}\frac{3}{x}e^{sinx^{3}}dx

We use substitution

Put x^{3}=t

We get x^{2}=\frac{t}{3}

\int_{1}^{64}\frac{e^{sint}}{t}dt=F(64-F(I)

\therefore K=64


Option 1)

16

This option is incorrect

Option 2)

63

This option is incorrect

Option 3)

64

This option is correct

Option 4)

15

This option is incorrect

Posted by

divya.saini

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