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Chapter 11: Problem 29

Find the indefinite integral and check your result by differentiation. $$ \int\left(x^{3}+2\right) d x $$

Short Answer

Expert verified
The indefinite integral of the function \(x^{3}+2\) is \( \frac{1}{4}x^4 + 2x + C \) and it has been verified by taking the derivative of the obtained result to return back the original function.

Step by step solution

01

Integrate Function

The integral of the given function \(x^{3}+2\) can be found by considering each component individually. The integral of \(x^{3}\) with respect to x is found by adding 1 to the power of x and then dividing by the new power. Similarly, the integral of constant, which is 2 here, is that constant times x. Therefore, \( \int (x^{3}+2) dx = \frac{1}{4}x^4 + 2x + C \). C is the constant of integration.
02

Check Result by Differentiation

To verify this result, take the derivative of the function \(\frac{1}{4}x^4 + 2x + C\) which should return the original function \(x^{3}+2\). The derivative is taken, again, component by component. Using the power rule, the derivative of \( \frac{1}{4}x^4 \) is \(x^{3}\), and the derivative of \(2x\) is \(2\). The derivative of the constant is zero. This gives \(x^{3}+2\), which is the same as the original function.

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