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Chapter 4: Problem 26

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(\frac{5}{t^{2}}+4 t^{2}\right) d t$$

Short Answer

Expert verified
Question: Find the indefinite integral of the function \(f(t) = \frac{5}{t^2} + 4t^2\). Answer: The indefinite integral of the function is \(\int\left(\frac{5}{t^{2}}+4 t^{2}\right) d t = -5t^{-1} + 4\frac{t^3}{3} + C\).

Step by step solution

01

Identify the terms of the integrand

In this case, the function that needs to be integrated is a sum of two terms: $$f(t) = \frac{5}{t^2} + 4t^2$$
02

Integrate each term separately

We will now integrate each term and then add up the results. The integral of a sum is equal to the sum of the integrals of its terms. For the first term, we have: $$\int\frac{5}{t^2}dt = 5\int\frac{1}{t^2}dt = 5\int t^{-2}dt$$ For the second term, we have: $$\int 4t^2 dt$$
03

Apply the Power Rule for Integration

The Power Rule for Integration states that for any real number 'n' not equal to -1, $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ Applying the Power Rule to the first term: $$5\int t^{-2}dt = 5\frac{t^{-1}}{-1} + C_1 = -5t^{-1} + C_1$$ Applying the Power Rule to the second term: $$\int 4t^2 dt = 4\frac{t^3}{3} + C_2$$
04

Add the individual integrals

Add the indefinite integrals from steps 3 and 4 together: $$\int\left(\frac{5}{t^{2}}+4 t^{2}\right) d t = -5t^{-1} + C_1 + 4\frac{t^3}{3} + C_2$$ Since we are combining constants, we can simply rewrite them as a single constant 'C': $$\int\left(\frac{5}{t^{2}}+4 t^{2}\right) d t = -5t^{-1} + 4\frac{t^3}{3} + C$$
05

Check the result by differentiation

Differentiate the obtained integral function to verify that it matches the original function. Using the Power Rule of differentiation, which is the reverse of the integration Power Rule: 1. Differentiate the first term, \(-5t^{-1}\): $$\frac{d(-5t^{-1})}{dt} = -5\cdot(-1)\cdot t^{-2} = \frac{5}{t^2}$$ 2. Differentiate the second term, \(4\frac{t^3}{3}\): $$\frac{d(4\frac{t^3}{3})}{dt} = 4\cdot 3\frac{t^2}{3} = 4t^2$$ Adding the derivatives of the first and second terms: $$\frac{5}{t^2} + 4t^2$$ Since the result of the differentiation matches the original function, the obtained integral is correct.

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