When finding a function given the antiderivative, a good approach is to take the derivative of . 

By using chain rule for the ln(4x)term, the derivative 1x is obtained. Power rule is used to differentiate the second term, and the last term, C, is simply a constant.

Therefore, the correct expression is f(x)=1x+3x2.

For this problem, keep in mind that both terms ( and sin) have inner and outer functions. This indicates that chain rule is involved.

Taking the derivative of  results in , so the second term in the expression must change from positive to negative. Lastly, the “C” term must also be included. Therefore, the correct expression for the antiderivative of  is .

The constant of integration, also known as , is used for indefinite integrals (in other words, the set of all possible antiderivatives of a function). This constant is used to communicate that on a connected domain, the indefinite integral is only defined up to an additive constant. 

Essentially, part of the function can be isolated by taking the antiderivative, but the positioning of this function may differ depending on what constants should be present in the equation. 

Because these constants could be either negative or positive, there are no restrictions on the exact sign of . Therefore, the correct answer is “Yes;  allows the expression of a general form of antiderivatives.”

Sometimes when there are multiple terms added or subtracted from one another within the antiderivative, it can be useful to invoke the following rule:

Separating the terms into two smaller antiderivative chunks can help declutter the problem at hand. Rewriting roots to look like exponents can also be useful.

The first chunk of this problem requires power rule, and the second is a trigonometry derivative identity. Finally, both  terms can be combined at the end to create a single constant of integration. Therefore, the correct answer is the following:

This problem looks complicated at first, but it is really just a trigonometry derivative identity with an inner function of . The inner function requires the application of chain rule.

Since , this identity is the starting point for this question.

First, find the general expression of the antiderivative. 

From here, the correct value of  must be identified in order to find the specific function of  from the general expression:

To finish this problem, we substitute  for  in the general expression:

To simplify this problem, it may be useful to split up the integral into more manageable  chunks: 

Another trick is to rewrite roots to look like exponents (in this case,). Then, proceed with taking the antiderivative, paying close attention to power rule:

The  and  terms can then be combined to create the total constant of integration. 

Finally, multiply constants through to identify the correct answer:

Remember that the notation for the antiderivative may appear as , but it may also take on a different form, as displayed in this question. 

Conceptually, it might be useful to think of the antiderivative as the “opposite” of the derivative function. Essentially, the antiderivative “undoes” the derivative.

However, since taking the derivative of a constant results in zero, any constants not accounted for by the antiderivative must be represented by the variable .

The expression that contains all the correct components of the antiderivative expression (including  and ), as well as acknowledges the relationship of , is .

To check our work, let’s take the derivative of our answer. This is a useful strategy to determine if the antiderivative was found correctly, since 

Note that there are two functions present ( and ), suggesting the need to use chain rule. When working with multiple functions in an antiderivative, a good strategy is to think through how chain rule would apply to the derivative, then implement the opposite steps to find the antiderivative.

We obtain the original function, . Notice how chain rule plays out - the derivative of the inner function, , is . Multiplying this derivative to  results in . Therefore, the correct antiderivative is .

To approach this problem, first find the general antiderivative expression for the function .

Then, the goal is to find the correct value of  that allows for the condition .

The value of Cis solved for by plugging in  into the newly found antiderivative formula. By substituting  into the antiderivative expression creates the specific antiderivative asked for in this question. Therefore, the antiderivative expression that satisfies the given requirements is .