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Chapter 4: Problem 58

Determine the following indefinite integrals. Check your work by differentiation. $$\int \frac{10 t^{5}-3}{t} d t$$

Short Answer

Expert verified
Question: Find the indefinite integral of the function $$\int \frac{10 t^{5}-3}{t} d t$$ and verify the result by differentiation. Solution: The indefinite integral of the given function is $$\int \frac{10 t^{5}-3}{t} d t = 2t^5 + 3\ln|t| + C$$, and its derivative verifies the result as follows: $$\frac{d}{dt} \left(2t^5 + 3\ln|t| + C\right) = 10t^4 - \frac{3}{t}$$

Step by step solution

01

Simplify the function in the integrand

Before integrating, we can simplify the function in the integrand: $$\frac{10 t^{5}-3}{t} = 10t^4 - \frac{3}{t}$$ Now, we can integrate the simplified function: $$\int \left(10t^4 - \frac{3}{t}\right) dt$$
02

Integrate the function

Integrate each term separately: $$\int 10t^4 dt - \int \frac{3}{t} dt$$ For the first term, we use the power rule for integration: $$\int t^n dt = \frac{t^{n+1}}{n+1} + C$$ For the second term, we know that: $$\int \frac{1}{t} dt = \ln |t| + C$$ So we have, $$\int 10t^ heightFor dt = 10 \cdot \frac{t^5}{5} + C = 2t^5 + C_1$$ $$\int \frac{3}{t} dt = 3 \cdot \ln |t| + C = 3\ln|t| + C_2$$ Now combine the integrals: $$\int \left(10t^4 - \frac{3}{t}\right) dt = 2t^5 + 3\ln|t| + C$$ where \(C = C_1 + C_2\)
03

Differentiate the result

To check our work, we differentiate the result: $$\frac{d}{dt} \left(2t^5 + 3\ln|t| + C\right)$$ Using the power rule for derivatives and the chain rule, we get: $$\frac{d}{dt}(2t^5) = 10t^4$$ $$\frac{d}{dt}(3\ln|t|) = \frac{3}{t}$$ Now, differentiate the constant: $$\frac{d}{dt}(C) = 0$$ Putting it all together, we have: $$\frac{d}{dt} \left(2t^5 + 3\ln|t| + C\right) = 10t^4 - \frac{3}{t}$$
04

Verify the result

We have found that the derivative of our integral is the original function in the integrand: $$\frac{d}{dt} \left(2t^5 + 3\ln|t| + C\right) = 10t^4 - \frac{3}{t}$$ Therefore, our integration and verification steps were successful. The indefinite integral of the given function is: $$\int \frac{10 t^{5}-3}{t} d t = 2t^5 + 3\ln|t| + C$$

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