Math Snap

STEP 1

1. The integral $\int_{1}^{2} \frac{1}{x^{5}} \, dx$ is a definite integral.
2. We will use the power rule for integration to solve this integral.

STEP 2

1. Rewrite the integrand in a simpler form.
2. Apply the power rule for integration.
3. Evaluate the antiderivative at the bounds.
4. Calculate the definite integral.

STEP 3

Rewrite the integrand $\frac{1}{x^{5}}$ in a simpler form using negative exponents:
$$\frac{1}{x^{5}} = x^{-5}$$

STEP 4

Apply the power rule for integration, which states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$:
$$\int x^{-5} \, dx = \frac{x^{-5+1}}{-5+1} + C = \frac{x^{-4}}{-4} + C = -\frac{1}{4}x^{-4} + C$$

STEP 5

Evaluate the antiderivative $-\frac{1}{4}x^{-4}$ at the bounds $1$ and $2$:
$$\left[-\frac{1}{4}x^{-4}\right]_{1}^{2} = \left(-\frac{1}{4}(2)^{-4}\right) - \left(-\frac{1}{4}(1)^{-4}\right)$$

STEP 6

Calculate the values:
$$-\frac{1}{4}(2)^{-4} = -\frac{1}{4} \cdot \frac{1}{16} = -\frac{1}{64}$$ $$-\frac{1}{4}(1)^{-4} = -\frac{1}{4} \cdot 1 = -\frac{1}{4}$$

Subtract the evaluated values to find the definite integral:
$$-\frac{1}{64} - \left(-\frac{1}{4}\right) = -\frac{1}{64} + \frac{1}{4}$$ Convert $\frac{1}{4}$ to a fraction with a denominator of 64:
$$\frac{1}{4} = \frac{16}{64}$$ Now perform the subtraction:
$$-\frac{1}{64} + \frac{16}{64} = \frac{15}{64}$$ The value of the definite integral is:
$$\boxed{\frac{15}{64}}$$