John W. answered • 07/20/20
Bachelor's Degree in Chemical Engineering
For the first question, the x can be distributed to get f(x) = 15x2 + 4x. Differentiation is a linear operation, which means that for differentiating sums of multiple terms, you can differentiate each term independently and add the results (there's a little more to the definition of a linear operation but I'll leave that off for the sake of keeping this answer at a reasonable length). In this case, we are finding the antiderivative, but we can still take advantage of this property. That is, if we find the antiderivative of 15x2 and the antiderivative of 4x, we can add those together to get the antiderivative of their sum. The antiderivative of 15x2 is 5x3 + C1, which we get using the Power Rule. We can also use the Power Rule to find the antiderivative of 4x, which is 2x2 + C2. Summing all the terms together, we find that the antiderivative of the function is 5x3 + 2x2 + C1 + C2, which can be simplified to 5x3 + 2x2 + C where C = C1 + C2. Note that the inclusion of the constant or constants in these expression is what makes it the "most general" antiderivative.
Now, let's differentiate the expression to check the answer. As mentioned earlier, differentiation is a linear operation and so we can differentiate each term individually and add the results.
The derivative of 5x3 is 15x2, which we get with the Power Rule.
The derivative of 2x2 is 4x, which we get with the Power Rule.
The derivative of C, a constant, is 0.
Adding the terms, we see that the derivative of 5x3 + 2x2 + C is 15x2 + 4x, which is our original function. Therefore, our antiderivative must be correct.
For the second question, I'm having a little trouble understanding the expression. I read it as:
f(x) = 7x3/2 - 5x2/3 + 2x-5 + (√4)x + 2/x
If I've misinterpreted the expression please let me know and I'll adjust my explanation accordingly.
Similarly to the last question, we can take the antiderivative of each term and add them to get an overall antiderivative.
The antiderivative of 7x3/2 is 7x5/2 / (5 / 2) + C1, or (14 / 5) * x5/2 + C1 (Power Rule).
The antiderivative of -5x2/3 is -5x5/3 / (5 / 3) + C2, or -3x5/3 + C2 (Power Rule).
The antiderivative of 2x-5 is 2x-4 / (-4) + C3, or (-1 / 2) * x-4 + C3 (Power Rule).
The expression (√4)x can be simplified to 2x, and the antiderivative of 2x is x2 + C4 (Power Rule).
The last antiderivative is a little different. 2/x = 2 * 1/x, and because derivatives are linear, if we find the antiderivative of 1/x, we can multiply it by 2 to get the antiderivative of 2 * 1/x. The derivative of ln|x| is 1/x, and so the antiderivative of 1/x is ln|x| + C. Therefore, the antiderivative of 2/x is 2 ln|x| + C5.
Summing them all up, we get (14 / 5) * x5/2 - 3x5/3 - (1 / 2) * x-4 + x2 + 2 ln|x| + C where
C = C1 + C2 + C3 + C4 + C5
Again we can check our answer by differentiating, and again we differentiate each term and sum them together:
The derivative of (14 / 5) * x5/2 is (14 / 5) * x3/2 * (5 / 2) or 7x3/2 (Power Rule).
The derivative of -3x5/3 is -3x2/3 * (5 / 3) or -5x2/3 (Power Rule).
The derivative of (-1 / 2) * x-4 is (-1 / 2) * x-5 * (-4) or 2x-5 (Power Rule).
The derivative of x2 is 2x (Power Rule).
The derivative of 2 ln|x| is 2 * (1/x) or 2/x.
The derivative of C, a constant, is 0.
Adding the terms, we see that the derivative of our answer is 7x3/2 - 5x2/3 + 2x-5 + 2x + 2/x, which is our original function. Therefore, our antiderivative must be correct.
Again please let me know if my interpretation of the expression was wrong, and I'll adjust my answer as soon as I can!
Also for any particularly nit-picky readers of this answer, all the derivatives and antiderivaties above are with respect to x.
