Applied Mathematics

Calculus

Easy

The value of the integral $\int_{-3}^{3} (x^3 - x) dx$ is :

$\frac{3}{2}$

$-\frac{1}{2}$

$\frac{x^3}{4} - \frac{x^2}{2} + C$

✅ Correct Option: 1

We need to find the value of $\int_{-3}^{3} (x^3 - x)\, dx$

Before integrating, let's check if $f(x) = x^3 - x$ is an odd function.

A function is odd if $f(-x) = -f(x)$ . If it is, then $\int_{-a}^{a} f(x)\, dx = 0$ because the areas on either side of the y-axis cancel out.

Replace $x$ with $-x$ :

$f(-x) = (-x)^3 - (-x)$

$= -x^3 + x$

$= -(x^3 - x)$

$= -f(x)$

Since $f(-x) = -f(x)$ , the function is odd.

Since $f(x) = x^3 - x$ is an odd function and we are integrating over the symmetric interval $[-3, 3]$ :

$\int_{-3}^{3} (x^3 - x)\, dx = 0$

The value of the integral is $\boxed{0}$

2023: 30 May Shift 3

If $C(x) = ax^2 - bx - c$ represents the total cost function then the slope of the tangent to the marginal cost curve at the point $(x, y)$ is :

2023: 22 May Shift 3

Match List - I with List - II.

	List - I		List - II
(A)	For break-even point	(I)	$< 0$
(B)	For maxima $\frac{d^2y}{dx^2}$	(II)	$\frac{dy}{dx} = 0$
(C)	For points of maxima/minima	(III)	$R(x) - C(x)$
(D)	$P(x) =$ Profit function	(IV)	$R(x) = C(x)$

Choose the correct answer from the options given below :

2022: 25 May Shift 1

If the cost function $C(x)$ of producing $x$ units of a commodity is given as $C(x) = x^3 - 60x^2 + 13x + 50$ , then the level of output for which the marginal cost is minimum is

2023: 11 Aug Shift 1

the general solution of the differential equation $(1 + y) dx - 2x dy = 0$ is :

2026: 14 May Shift 2

The second order derivative of the function $y=\log(4x^3+6)$ with respect to 'x' will be:

2023: 15 June Shift 2

A product costs the manufacturer ₹ 20 per unit. The demand function is given by $p(x) = 1000 - 20x$ , then the quantity for maximum profit is :

The value of the integral ∫−33(x3−x)dx\int_{-3}^{3} (x^3 - x) dx∫−33​(x3−x)dx is :

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

The value of the integral $\int_{-3}^{3} (x^3 - x) dx$ is :