Question

Need help question #1. Show steps please

Answer 1

Answer:

C

Step-by-step explanation:

We want to integrate:

$\displaystyle \int\frac{4x^4+3}{4x^5+15x+2}\,dx$

Notice that the expression in the denominator is quite similar to the expression in the numerator. So, we can try performing u-substitution. Let u be the function in the denominator. So:

$u=4x^5+15x+2$

By differentiating both sides with respect to x:

$\displaystyle \frac{du}{dx}=20x^4+15$

We can "multiply" both sides by dx:

$du=20x^4+15\,dx$

And divide both sides by 5:

$\displaystyle \frac{1}{5}\, du=4x^4+3\,dx$

Rewriting our original integral yields:

$\displaystyle \int \frac{1}{4x^5+15x+2}(4x^4+3\, dx)$

Substitute:

$\displaystyle =\int \frac{1}{u}\Big(\frac{1}{5} \, du\Big)$

Simplify:

$\displaystyle =\frac{1}{5}\int \frac{1}{u}\, du$

This is a common integral:

$\displaystyle =\frac{1}{5}\ln|u|$

Back-substitute. Of course, we need the constant of integration:

$\displaystyle =\frac{1}{5}\ln|4x^5+15x+2|+C$

Our answer is C.