Indefinite Integral Calculation

Hi Yasemin, let's solve this integration problem together. We are asked to evaluate an indefinite integral, but the full expression is partly cut off. Based on the options, the integral likely looks like this.

When we see a complex fraction where the derivative of the denominator resembles the numerator, the method of substitution is the best approach.

Let's set u equal to the term inside the cube on the bottom: x cubed plus three x minus five.

Now, let's differentiate both sides with respect to x. The derivative of x cubed is three x squared, and the derivative of three x is three.

Notice that we can factor out a three from the right side to get three times the quantity x squared plus one times d x.

If we divide both sides by three, we find that one third d u is equal to x squared plus one times d x, which matches our numerator perfectly.

Now, let's substitute these values back into our original integral.