Let frac{d}{dx}F(x) = frac{e^{sin x}}{x} for | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given that $\frac{d}{dx}F(x) = \frac{e^{\sin x}}{x}$.We need to evaluate the integral $I = \int_{1}^{4} \frac{3}{x} e^{\sin(x^3)} dx$.Multiply the

Vedclass · Accepted Answer

$64$

Vedclass · Answer

(D) Given that $\frac{d}{dx}F(x) = \frac{e^{\sin x}}{x}$.We need to evaluate the integral $I = \int_{1}^{4} \frac{3}{x} e^{\sin(x^3)} dx$.Multiply the numerator and denominator by $x^2$ to facilitate substitution:$I = \int_{1}^{4} \frac{3x^2}{x^3} e^{\sin(x^3)} dx$.Let $t = x^3$,then $dt = 3x^2 dx$.When $x = 1$,$t = 1^3 = 1$.When $x = 4$,$t = 4^3 = 64$.Substituting these into the integral:$I = \int_{1}^{64} \frac{e^{\sin t}}{t} dt$.Since $\frac{d}{dt}F(t) = \frac{e^{\sin t}}{t}$,the integral becomes:$I = [F(t)]_{1}^{64} = F(64) - F(1)$.Comparing this with the given expression $F(k) - F(1)$,we get $k = 64$.

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

Similar Questions

$\int_0^{\pi / 4} \frac{\sec x}{3 \cos x+4 \sin x} d x=$

The value of $\int_1^{e^2} \frac{dx}{x(1 + \ln x)^2}$ is

The value of the integral $\int_0^{\log 5} {\frac{{{e^x}\sqrt {{e^x} - 1} }}{{{e^x} + 3}}} \,dx = $

Let $\alpha$ and $\beta$ $(\alpha < \beta)$ be the roots of $18x^2 - 9\pi x + \pi^2 = 0$,$f(x) = x^2$,and $g(x) = \cos x$. Then $\int_{\alpha}^{\beta} x (g \circ f(x)) dx =$

Evaluate the definite integral $\int_{2}^{3} \frac{x}{x^{2}+1} dx$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module