The integral evaluates to [0.5x^4 - x^3 + 4x] from 1 to 2 = (8 - 8 + 8) - (0.5 - 1 + 4) = 8 - 2.5 = 5.5.

The integral evaluates to 20.

Evaluating the integral gives [x^3 + x^2]_0^1 = (1 + 1) - (0 + 0) = 2.

Using L'Hôpital's rule, the limit evaluates to 5.

Using L'Hôpital's Rule, the limit evaluates to 2.

The limit lim(x->0) (sin(x)/x) = 1.

Using L'Hôpital's rule, the limit evaluates to 5.

The limit lim(x→0) (sin(x)/x) = 1.

As x approaches infinity, 1/x approaches 0.

The principal quantum number n for the f subshell is 4.

The principal quantum number (n) indicates the energy level of the electron. For 4s, n=4.

In the ground state, the electron is in the first energy level, which corresponds to n=1.

Sodium has an atomic number of 11, and its electron configuration is 1s² 2s² 2p⁶ 3s¹. The outermost electrons are in n=3.

Potassium (K) has an atomic number of 19, and its electron configuration ends in 4s, so the outermost electron has n=4.

Potassium has the electronic configuration [Ar] 4s1, so the outermost electron is in n=4.

The ground state of hydrogen corresponds to n=1.

Potassium has an atomic number of 19, and its outermost electrons are in the n=4 shell.

Sodium has an atomic number of 11, and its outermost electrons are in the n=3 shell.

The series diverges to infinity as n approaches infinity.

The term containing x^5 is C(8,5)(1/2)^3 = 56 * 1/8 = 7.

The term containing x^5 is C(8,5)(2)^3 = 56 * 8 = 448.

The universal gas constant R is 8.314 J/(mol·K).

The universal gas constant R is 0.0821 L·atm/(K·mol).

Subtract 3 from both sides: 2x = 8. Then divide by 2: x = 4.

Solving for x: 3x = 21 => x = 7.

Using the quadratic formula, x = [8 ± sqrt(64 - 48)] / 4 = [8 ± 4] / 4, giving x = 3 or x = 1.

Solving for x: 3x = 12, thus x = 4.

Expanding gives 5x - 10 = 3x + 4. Rearranging gives 2x = 14, thus x = 7.

Rearranging gives 5x - 2x = 12 + 3 => 3x = 15 => x = 5.

Rearranging gives 5x - 2x = 6 + 3 => 3x = 9 => x = 3.