Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral and check your result by differentiation. $$ \int \frac{1}{4 x^{2}} d x $$

Step-by-Step Solution

Answer

The integral of \( \frac{1}{4 x^{2}} dx \) is \( -\frac{1}{4x}\)

Step 1: Write the integral in power form

Rewrite the integral in power form to make it easy to use the power rule. Here, \( \frac{1}{4 x^{2}} \) can be written as \( \frac{1}{4} x^{-2} \). So, the integral becomes \( \int \frac{1}{4} x^{-2} dx\).

Step 2: Evaluate the integral using the power rule

According to the power rule, \(\int x^{n} dx = \frac{1}{n+1} x^{n+1} + c\). Thus, the integral of \( \frac{1}{4} x^{-2} \) would be \( \frac{1}{4*(-2 + 1)} x^{-2+1} + c\). Simplifying the expression, we get \( -\frac{1}{4} x^{-1} + c \) or \( -\frac{1}{4x}\).

Step 3: Verify the integral by differentiation

To verify the result, differentiate the obtained integral \( -\frac{1}{4x}\). The derivative of \(-\frac{1}{4x}\) using the power rule of differentiation is \(\frac{1}{4 x^{2}}\), which is the original integrand, so this verifies the integrated solution.