Explanation

Let u = 5x, du/dx = 5, du = 5dx, dx = du/5

∫sin5xdx = (1/5)∫sinu du

= (1/5)(-cosu) + c

= -1/5cos5x + c

Explanation

The equation is separable:

  dy/dx = –y

  ⇒ dy/y = –dx

  ⇒ ∫(1/y) dy = ∫(–1) dx

  ⇒ ln|y| = –x + C

  ⇒ y = Ce⁻ˣ (where C is the constant of integration)

Explanation

The number of ways to arrange n distinct objects in a circular ring is (n-1)!.

For 5 keys, the number of ways = (5-1)! = 4! = 24.

Explanation

The sum of the first n odd numbers is given by:

1 + 3 + 5 + ... + (2n - 1) = n²

Explanation

tan θ < 0: Quadrants II and IV

sin θ < 0: Quadrants III and IV

Common quadrant: IV

Explanation

The expansion of (1 + x)⁵ can be found using the Binomial Theorem.

(1 + x)⁵ = 1 + 5x + 10x² + 10x³ + 5x⁴ + x⁵

Sum of coefficients = 1 + 5 + 10 + 10 + 5 + 1 = 32

Alternatively, we can find the sum by substituting x = 1:

(1 + 1)⁵ = 2⁵ = 32

Explanation

Let's say the two numbers are x and y.

Given:

x + y = 24 ... (1)

x - y = 6 ... (2)

Adding (1) and (2):

2x = 30

x = 15

Substituting x in (1):

15 + y = 24

y = 9

The numbers are 15 and 9.

Explanation

To find the square of the distance between two points $A(x_1, y_1)$ and B(x2​,y2​):

AB^2=(x2​−x1​)^2+(y2​−y1​)^2 $$= (3 - 4)^2 + (5 - (-1))^2 = (-1)^2 + (6)^2 = 1 + 36 = 37$$

=(−1)^2+(6)^2

=1+36

$$=37$$

Explanation

Given the points (x + 5, 8) and (9, y - 6) are equal, we can set up the equations:

x + 5 = 9 ... (1)

8 = y - 6 ... (2)

Solving (1):

x = 9 - 5

x = 4

Solving (2):

y = 8 + 6

y = 14

The values of x and y are x = 4 and y = 14.

Explanation

The given statement:

If a : b = c : d then a : c = b : d

This is known as the Alternendo theorem.

The Alternendo theorem states that if a/b = c/d, then a/c = b/d.