Explanation

Explanation

Given that the two sides of the isosceles triangle are equal:

3x - 2y + 2 = x - y + 7

This simplifies to:

2x - y = 5

The sum of the two equal sides is 12 cm:

(3x - 2y + 2) + (x - y + 7) = 12

This simplifies to:

4x - 3y = 3

Solving these two equations simultaneously, we find x = 6.

Explanation

$$lim_{x to 0} frac{1 - cos x}{x^2} = lim_{x to 0} frac{sin x}{2x} = frac{1}{2}$$

Explanation

What is the value of the integral:

∫e^x2+lnx dx

Step 1: Simplify the integrand

Recall:

e^a+b=e^a⋅e^b

So,

e^x^2+lnx=e^x^2⋅e^lnx=x⋅e^x^2

Step 2: The integral becomes:

∫x⋅e^x^2 dx

Now use substitution:

Let $u = x^2 Rightarrow du = 2x , dx Rightarrow frac{1}{2} du = x , dx$

So,

$$int x cdot e^{x^2} , dx = frac{1}{2} int e^u , du = frac{1}{2} e^u + C = frac{1}{2} e^{x^2} + C$$

=1​/2∫e^u du

=1​/2e^u+C

=1​/2ex^2 +C

Explanation

We use the product rule since it's the product of two functions:

$$f(x) = x^3 cdot ln(x)$$

Let:

• $u = x^3 Rightarrow u' = 3x^2$

• $v = ln(x) Rightarrow v' = frac{1}{x}$

Apply the product rule:

f′(x)=u′v+uv′=3x^2⋅ln(x)+x^3⋅1/x​
f′(x)=3^x2ln(x)+x^2

Explanation

∫01​(2x+1)dx

Step 1: Integrate the function

We integrate each term separately:

$$int (2x + 1), dx = int 2x, dx + int 1, dx = x^2 + x + C$$

Step 2: Apply the limits from 0 to 1

$$left[ x^2 + x right]_0^1 = (1^2 + 1) - (0^2 + 0) = 1 + 1 = 2$$

Explanation

This is a product of two functions, so we apply the product rule:

d​[u⋅v]/dx=u′⋅v+u⋅v′

Here,

• u=x^2, sou′=2x

• $v = sin(x)$, so $v' = cos(x)$

Now apply the rule:

$$f'(x) = 2x cdot sin(x) + x^2 cdot cos(x)$$

Explanation

To integrate $x^2$, use the power rule of integration:

$$int x^n dx = frac{x^{n+1}}{n+1} + C$$

Here, $n = 2$, so:

$$int x^2 dx = frac{x^{2+1}}{2+1} + C = frac{x^3}{3} + C$$2+1/2+1​+C

=x^3​/3+C

Explanation

  f(x) = |x| has a cusp at x = 0, making it non-differentiable there.

A function is differentiable at a point if its derivative exists at that point. 

The absolute value function, (f(x)=|x|), has a sharp corner at (x=0). 

Functions with sharp corners are not differentiable at those points.

Explanation

The formula ∑x/n calculates the arithmetic mean of a set of numbers, where:

∑x = sum of all values

n = number of values

This formula provides the average value of the dataset.