The value of int_{-pi/2}^{pi/2} (3sin x + sin | vedclass

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

Question

(C) Let $I = \int_{-\pi/2}^{\pi/2} (3\sin x + \sin^3 x) \, dx$.Consider the function $f(x) = 3\sin x + \sin^3 x$.Check if the function is even or odd

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(C) Let $I = \int_{-\pi/2}^{\pi/2} (3\sin x + \sin^3 x) \, dx$.Consider the function $f(x) = 3\sin x + \sin^3 x$.Check if the function is even or odd by substituting $-x$ for $x$:$f(-x) = 3\sin(-x) + \sin^3(-x)$Since $\sin(-x) = -\sin x$,we have:$f(-x) = 3(-\sin x) + (-\sin x)^3 = -3\sin x - \sin^3 x = -(3\sin x + \sin^3 x) = -f(x)$.Since $f(-x) = -f(x)$,the function $f(x)$ is an odd function.According to the property of definite integrals,if $f(x)$ is an odd function,then $\int_{-a}^{a} f(x) \, dx = 0$.Therefore,$\int_{-\pi/2}^{\pi/2} (3\sin x + \sin^3 x) \, dx = 0$.

The value of $\int_{-\pi/2}^{\pi/2} (3\sin x + \sin^3 x) \, dx$ is

Similar Questions

If $\int_0^{2a} f(x) \, dx = 2 \int_0^a f(x) \, dx$,then:

Let $T > 0$ be a fixed number. Suppose $f$ is a continuous function such that for all $x \in \mathbb{R}, f(x + T) = f(x)$. If $I = \int_{0}^{T} f(x) dx$,then the value of $\int_{3}^{3 + 3T} f(2x) dx$ is

$\int_{2}^{4} \frac{\log(x^2)}{\log(x^2) + \log(36 - 12x + x^2)} \, dx = $

The value of the integral $I = \int_{1/2014}^{2014} \frac{\tan^{-1} x}{x} dx$ is

Suppose $f$ is such that $f(-x) = -f(x)$ for every real $x$ and $\int_{0}^{1} f(x) dx = 5$,then $\int_{-1}^{0} f(t) dt = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module