Question

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

Short answer

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

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.

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.