Finding Indefinite Integrals

In Exercises 17–...

Question

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(1/x² − x² − 1/3) dx

Accepted Answer

Welcome back everyone. Evaluate the indefinite integral 3 x 2 minus 1 divided by x cubed + 4 multiplied by the x. For this problem, what we can do is simply rewrite our integrals 3 x 2 minus using the reciprocal rule, we can rewrite 1 divided by x cubed as x raised to the power of -3. Plus 4 d x. Now, let's split our integral into 3 parts. For the first part, we're going to integrate x squared dx, and we're going to factor out 3 because it's a constant. Then we're going to subtract the integral of x to the power of -3 d x, and then we are going to integrate dx, right, because we can factor out 4, which is a constant again. And from here we can notice that all that we have to do is apply the power rule. For the first integral, we're going to get 3 multiplied by x to the power of 2 + 1 divided by 2 + 1. Minus for the second integral that would be x to the power of -3 + 1 divided by -3 + 1 plus for the final integral we get 4 multiplied by x. That's the integral of d x right plus a constant of integration c. And now all that we have to do is simplify. So that would be 3 divided by 3 x cubed. Minus. Now notice that we're subtracting a negative numbers, we can just say plus. X to the power of -2 divided by now-2 becomes 2, right? Plus 4 x plus c simplifying further 3 divided by 3 is simply 1, so we get x cubed plus 1 divided by 2 x squared using the reciprocal rule. Plus 4 x plus c. That will be our final answer and thank you for watching.

Finding Indefinite IntegralsIn Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.∫(1/x² − x² − 1/3) dx

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Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(1/x² − x² − 1/3) dx