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

$\int_0^{\frac{\pi}{2}} \frac{d x}{\cos x-\sqrt{3} \sin x}=$

The function $L(x) = \int_1^x \frac{dt}{t}$ satisfies the equation

Evaluate the definite integral $\int_{0}^{1} \frac{d x}{1+x^{2}}$.

The value of the integral $\int \limits_{-\log _{e} 2}^{\log _e 2} e^x \ln \left(e^x+\sqrt{1+e^{2 x}}\right) d x$ is equal to

$\int_0^1 \sin \left(2 \tan ^{-1} \sqrt{\frac{1+x}{1-x}}\right) d x$ is equal to :

Vedclass Test Series

Exam Paper Generator

Online Exam Module