23–68. Indefinite integrals Determine the following indefinite...

Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (3/x⁴ + 2 - 3/x²)

Accepted Answer

Welcome back, everyone. Evaluate the indefinite inag. I of 4 divided by X the power of 5 plus 5 minus 2 divided by X cubed DX. So first of all, what we can do in this problem is apply the reciprocal rule. We have the integral of 4 multiplied by 1 divided by X to the power of 5 + 5 minus 2 multiplied by 1 divided by X cubed. And now applying the reciprocal rule, we can express 1 divided by X to the power of 5 as. X to the power of -5. And one divided by X cubed as X er raises the power of -3. Now, the reason why this is useful is simply because we can apply the power rule for integration. Now, let's split our integral into 3 parts and factor out the constants. So the first integral is going to be 4 multiplied by the integral. Of X raised the power of -5D X. The second in a role is going to be 5 multiplied by. The in a role of D X we have factored out 5, and the final one is going to be -2 multiplied by the in a role of X to the power of -3. And now we are ready to apply the power rule. For the first in a role, we get X to the power of -5 + 1, that's -4 divided by -4. So we add 1 to the original exponent and divide the result by the new exponent. So we get -4 and divide by -4. For the second one we get 5 multiplied by the inaugural of T X which is X. And for the final one we subtract 2 multiplied by the integral of X, raise the power of -3, which is X. Raise the power of -3 + 1. That's -2 divided by the new exponent -2. And finally we have to add a constant of integration C. So now let's value the result. For the first part we can simplify, for divided by -4 is -1, so we get negative. And X raises the power of -4 can be written as 1 divided by X to the power of 4 applying the reciprocal rule. We then add 5 X. And we subtract 2 divided by -2, we can cancel those out. And negative divided by negative gives us positive, right, meaning we add. And x raises the power of -2 can be written as 1 divided by X squared using the same reciprocal rule. And finally we add a constant of integration C. Well then we have got our final answer and thank you for watching.

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (3/x⁴ + 2 - 3/x²)

Key Concepts

Indefinite Integrals

Integration Techniques

Verification by Differentiation

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