Problem 58 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Find the indefinite integral of the function: $$\int \frac{10t^5 - 3}{t} dt$$ Answer: The indefinite integral of the given function is: $$2t^5 + 3\ln{|t|} + C$$

Step 1: Simplify Integrand

Simplify each integrand in the separate integrals: $$10\int t^{5-1} dt - 3 \int \frac{1}{t} dt$$ Now we have: $$10\int t^4 dt - 3\int \frac{1}{t} dt$$

Step 2: Integrate Each Term Separately

Perform the integration of each term separately: $$10\int t^4 dt = 10 \cdot \frac{t^5}{5} = 2t^5$$ $$3\int \frac{1}{t} dt = 3\ln{|t|}$$ Now combine the two results: $$2t^5 + 3\ln{|t|} + C$$ Where C is the integration constant.

Step 3: Verify the Result by Differentiation

Now, let's differentiate the result with respect to t to check our work: $$\frac{d}{dt}(2t^5 + 3\ln{|t|} + C) = 10t^4 - 3\frac{1}{t}$$ The differentiation result matches the integrand, confirming our result. Thus, the solution to the indefinite integral is: $$\int \frac{10t^5 - 3}{t} dt = 2t^5 + 3\ln{|t|} + C$$