Suppose that F(x) is an antiderivative of f(x | vedclass

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

Question

(A) Given that $F(x)$ is an antiderivative of $f(x) = \frac{\sin x}{x}$,we have $\int \frac{\sin x}{x} dx = F(x) + C$. We need to evaluate the integra

Vedclass · Accepted Answer

$F(6) - F(2)$

Vedclass · Answer

(A) Given that $F(x)$ is an antiderivative of $f(x) = \frac{\sin x}{x}$,we have $\int \frac{\sin x}{x} dx = F(x) + C$. We need to evaluate the integral $I = \int_{1}^{3} \frac{\sin 2x}{x} dx$. Let $t = 2x$,then $dt = 2 dx$,which implies $dx = \frac{dt}{2}$. When $x = 1$,$t = 2$. When $x = 3$,$t = 6$. Substituting these into the integral: $I = \int_{2}^{6} \frac{\sin t}{t/2} \cdot \frac{dt}{2} = \int_{2}^{6} \frac{\sin t}{t} dt$. Since $\int \frac{\sin t}{t} dt = F(t)$,by the Fundamental Theorem of Calculus,$\int_{2}^{6} \frac{\sin t}{t} dt = F(6) - F(2)$. Thus,the correct option is $A$.

Suppose that $F(x)$ is an antiderivative of $f(x) = \frac{\sin x}{x}$,$x > 0$. Then $\int_{1}^{3} \frac{\sin 2x}{x} dx$ can be expressed as:

Similar Questions

The integral $\int_0^1 \cot^{-1}(1 + x + x^2) dx$ is equal to:

$\int_0^{\pi / 4} \tan ^2(x) \, dx =$

Let $f(x) = \max \{x + |x|, x - [x]\}$, where $[x]$ stands for the greatest integer not greater than $x$. Then $\int_{-3}^3 f(x) \, dx$ has the value:

The approximate value of $\int_{1}^{9} x^2 dx$ by using the trapezoidal rule with $4$ equal intervals is:

The value of $\int_0^1 {x^2 e^x dx}$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module