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Chapter 11: Problem 35

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

Short Answer

Expert verified
The indefinite integral of \(1/x^4\) is \(-\frac{1}{3}x^{-3} + C\), where C represents the constant of integration.

Step by step solution

01

Apply the power rule for integration

To integrate \(1/x^4\), write the integral as \(\int x^{-4} dx\) and apply the power rule. The answer is \(x^{-4+1}/(-4+1) + C\) which simplifies to \(-\frac{1}{3}x^{-3} + C\).
02

Confirmation by differentiation

Now, check the result by differentiation using the power rule: the derivative of a constant term vanishes and the derivative of \(-\frac{1}{3}x^{-3}\) is \( \frac{1}{x^4}\). This confirms the result of the integration.

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