We know that is a constant so we can just integrate this term regularly.

We see the exponential term needs to be integrated using exponential integration methods. In this case we will use the standard

Integrate

To integrate this function we can use the reverse chain rule. To do so we must manipulate the integral into the form,

Therefore, we multiply the integral by

Now the integral is in the form , except multiplied by the constant .

Note: We can see from the above expression that we can quickly integrate exponentials in this specific form using the formula (this is sometimes known as the standard form for integrating exponentials),

We know that the integral of any exponential is given by:

therefore:

Therefore, the solution to this is (A)

Step 1 - Factor out from the integral (we are going to use standard exponential integration as opposed to the reverse chain rule)

Step 2 - Integrate using standard exponential integration

Step 3 - Evaluate the bounds

Step 1 - Move the denominator to the numerator using index laws,

Since , we can see that .

Therefore the function is in the chain rule form, where , and ,

Step 2 - Integrate

Step 3 - Evaluate bounds

Step 1 - Split the fraction

Step 2 - Apply index laws to simplify the expression

Step 3 - Integrate

Step 1 - Write out the full integral

Step 2 - Pull out the constant from the integral

Step 3 - Integrate

Step 4 - Apply the bounds and simplify

Therefore,

This function is special because it is an odd function, as the integral of the function from equidistant bounds from the origin (i.e from to ) is equal to .

Step 1 - Integrate the function from to

Note: is the volume of tea that Saivan drinks, and is the time in minutes.

Step 2 - Evaluate the integral

Step 1 - Make the subject

Step 2 - Integrate both sides

Where is just a constant, which we will find in the next step

Step 3 - Substitute in the point into the equation to solve for

Step 4 - Write write the function completely