Explanation

Step 1: Evaluate the integral

∫(x² - x - 1) dx = (x³ / 3) - (x² / 2) - x

Step 2: Apply the limits of integration

[(1³ / 3) - (1² / 2) - 1] - [((-1)³ / 3) - ((-1)² / 2) - (-1)]

= [(1/3) - (1/2) - 1] - [(-1/3) - (1/2) + 1]

Step 3: Simplify the expression

= (1/3) - (1/2) - 1 + (1/3) + (1/2) - 1

= (2/3) - 2

= (2/3) - (6/3)

= -4/3

Explanation

Total ways to draw 3 balls = C(15, 3) = 455

Ways to draw 3 red balls = C(5, 3) = 10

Probability = 10/455 = 2/91

Explanation

Harmonic Mean (H.M): The reciprocal of the arithmetic mean of the reciprocals of the numbers.

Geometric Mean (G.M): The nth root of the product of the numbers.

Arithmetic Mean (A.M): The sum of the numbers divided by the count of numbers.

Explanation

A compound proposition that is always true, regardless of the truth values of its components, is called a tautology.

The statement "P ∨ ¬P" (P or not P) is a tautology because it is always true, whether P is true or false.

Explanation

1. Intersection of $D_f$ and $D_g$: The function $\frac{f}{g}$ is defined only where both $f$ and $g$ are defined.

2. Excluding zeros of $g$: The function $\frac{f}{g}$ is undefined where $g(x)=0$ because division by zero is not allowed.

Explanation

Eliminate means ختم کریں

Explanation

Since f(x) = x^2 is always non-negative, there is no real x such that f(x) = -9.

So, the inverse image of -9 is the empty set:

f^(-1)(-9) = Ø

Explanation

Given y = acos(x) + bsin(x) + 4

First derivative: dy/dx = -asin(x) + bcos(x)

Second derivative: d²y/dx² = -acos(x) - bsin(x)

Notice that d²y/dx² = -(acos(x) + bsin(x))

Given y = acos(x) + bsin(x) + 4, we rearrange to find acos(x) + bsin(x) = y - 4

Substituting into the second derivative equation gives:

d²y/dx² = -(y - 4)

d²y/dx² = -y + 4

d²y/dx² + y = 4

The answer is d²y/dx² + y = 4.

Explanation

Simply Connected: A region is simply connected if every closed curve within it can be continuously shrunk to a point without leaving the region.

Multi-Connected: A region is multi-connected if it is not simply connected.

This meaning it has "holes" or other topological features that prevent some closed curves from being shrunk to a point within the region.

Explanation

1. Fundamental Theorem of Calculus:
   If $f$ is continuous on $[a,b]$ and $F$ is any antiderivative of $f$ on $[a,b]$, then:

   $${<b><font\; color="\#0000ff">\int </font></b>}_{\mathrm{<b><font\; color="\#0000ff">a</font></b>}}^{\mathrm{<b><font\; color="\#0000ff">b</font></b>}}\mathrm{<b><font\; color="\#0000ff">f</font></b>}<b><font\; color="\#0000ff">(</font></b>\mathrm{<b><font\; color="\#0000ff">x</font></b>}<b><font\; color="\#0000ff">)</font></b>\text{<b><font color="\#0000ff">\hspace{0.17em}</font></b>}\mathrm{<b><font\; color="\#0000ff">d</font></b>}\mathrm{<b><font\; color="\#0000ff">x</font></b>}<b><font\; color="\#0000ff">=</font></b>\mathrm{<b><font\; color="\#0000ff">F</font></b>}<b><font\; color="\#0000ff">(</font></b>\mathrm{<b><font\; color="\#0000ff">b</font></b>}<b><font\; color="\#0000ff">)</font></b><b><font\; color="\#0000ff">-</font></b>\mathrm{<b><font\; color="\#0000ff">F</font></b>}<b><font\; color="\#0000ff">(</font></b>\mathrm{<b><font\; color="\#0000ff">a</font></b>}<b><font\; color="\#0000ff">)</font></b>$$