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Chapter 4: Problem 24

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(3 u^{-2}-4 u^{2}+1\right) d u$$

Short Answer

Expert verified
Question: Determine the indefinite integral of the function \(3u^{-2}-4u^2+1\). Answer: The indefinite integral of the given function is \(-3u^{-1} - \frac{4}{3}u^3 + u + C\).

Step by step solution

01

Identify the individual terms

The given function is \((3u^{-2}-4u^2+1)\). To find the integral, we need to integrate each term separately. We will do this according to the power rule for integration.
02

Integrate each term

Now, apply the power rule to each individual term: \(\int 3u^{-2} du = 3 \int u^{-2} du\), \(\int -4u^2 du = -4 \int u^2 du\), and \(\int 1 du = 1 \int du\)
03

Apply the power rule to the integrals

When applying the power rule, we get the following results: For \(\int 3u^{-2}du\): Add \(1\) to the exponent: \(-2 + 1 = -1\), then divide by the revised exponent: $$3 \frac{u^{-1}}{-1} = -3u^{-1}$$ For \(\int -4u^2du\): Add \(1\) to the exponent: \(2 + 1 = 3\), then divide by the revised exponent: $$-4 \frac{u^{3}}{3} = -\frac{4}{3}u^3$$ For \(\int 1 du\): Since it's just a constant, the integral is simply \(u\): $$1 \int du = u$$
04

Combine all terms and add constant of integration

Now we combine all the terms and, since we are doing indefinite integration, we must add a constant of integration, C: $$-3u^{-1} -\frac{4}{3}u^3 + u + C$$
05

Verify the result by differentiation

Differentiate our answer with respect to \(u\) to make sure it matches the original function: $$\frac{d}{du}(-3u^{-1}-\frac{4}{3}u^3+u+C) = 3u^{-2} -4u^2 + 1$$ The results match, so our integral and verification are correct. The final answer for the indefinite integral is: $$\int (3u^{-2}-4u^2+1) du = -3u^{-1} - \frac{4}{3}u^3 + u + C$$

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