Evaluate $\int\left(\frac{5}{x^2}+6 x\right) d x$
Answer (1)
Rewrite the integrand: ∫ ( x 2 5 + 6 x ) d x = ∫ ( 5 x − 2 + 6 x ) d x .
Apply the power rule for integration: ∫ x n d x = n + 1 x n + 1 + C .
Integrate each term separately: ∫ 5 x − 2 d x = − x 5 and ∫ 6 x d x = 3 x 2 .
Combine the results and add the constant of integration: 3 x 2 − x 5 + C .
Explanation
Problem Analysis We are asked to evaluate the indefinite integral of the function x 2 5 + 6 x . This means we need to find a function whose derivative is x 2 5 + 6 x .
Applying the Power Rule We can rewrite the integral as ∫ ( 5 x − 2 + 6 x ) d x . Now, we can apply the power rule for integration to each term. The power rule states that ∫ x n d x = n + 1 x n + 1 + C , where n = − 1 and C is the constant of integration.
Integrating the First Term First, let's integrate 5 x − 2 with respect to x : ∫ 5 x − 2 d x = 5 ∫ x − 2 d x = 5 ⋅ − 2 + 1 x − 2 + 1 = 5 ⋅ − 1 x − 1 = − 5 x − 1 = − x 5 .
Integrating the Second Term Next, let's integrate 6 x with respect to x : ∫ 6 x d x = 6 ∫ x d x = 6 ⋅ 1 + 1 x 1 + 1 = 6 ⋅ 2 x 2 = 3 x 2 .
Combining the Results Now, we combine the results and add the constant of integration C : ∫ ( x 2 5 + 6 x ) d x = − x 5 + 3 x 2 + C .
Final Answer Therefore, the indefinite integral of x 2 5 + 6 x is 3 x 2 − x 5 + C .
Examples
Imagine you are calculating the total cost of a project where the cost rate varies with time. If the cost rate is given by a function like t 2 5 + 6 t , where t is time, integrating this function will give you the total cost as a function of time. This allows you to predict the total cost at any given time during the project.
