Questions: Evaluate the following indefinite integral. [ int frac14 x^4 d x ]

$Evaluate the following indefinite integral. [ int frac14 x^4 d x ]$

Transcript text: Evaluate the following indefinite integral. \[ \int \frac{1}{4 x^{4}} d x \]

Solution

Step 1: Rewrite the Integrand

We start with the integral

$\int \frac{1}{4 x^{4}} \, dx.$

This can be rewritten as

$\int \frac{1}{4} x^{-4} \, dx.$

Step 2: Apply the Power Rule

Using the power rule for integration, we have

$\int x^{-4} \, dx = \frac{x^{-3}}{-3} + C = -\frac{1}{3 x^{3}} + C.$

Thus, incorporating the constant factor $\frac{1}{4}$ , we get

$\int \frac{1}{4} x^{-4} \, dx = \frac{1}{4} \left(-\frac{1}{3 x^{3}} + C\right) = -\frac{1}{12 x^{3}} + C.$

The result of the indefinite integral is

$\boxed{-\frac{1}{12 x^{3}} + C}.$

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