Question

The integral $\int \frac{2x^{12}+5x^9}{(x^5+x^3+1)^3} dx$ is equal to

• A. $\frac{-x^{10}}{2(x^5+x^3+1)^2} + C$
• B. $\frac{-x^5}{(x^5+x^3+1)^2} + C$
• C. $\frac{x^{10}}{2(x^5+x^3+1)^2} + C$
• D. $\frac{x^5}{2(x^5+x^3+1)^2} + C$

Answer 1

Answer: (C)

$\frac{x^{10}}{2(x^5+x^3+1)^2} + C$

Answer 2

To evaluate the integral $I = \int \frac{2x^{12}+5x^9}{(x^5+x^3+1)^3} dx$, we can use a substitution method.

Step 1: Manipulate the integrand

The denominator is $(x^5+x^3+1)^3$. A common strategy for such integrals is to divide both the numerator and the denominator by the highest power of $x$ inside the parenthesis, raised to the power of the exponent outside. In this case, the highest power of $x$ inside is $x^5$, and the exponent outside is $3$. So, we divide by $(x^5)^3 = x^{15}$.

Divide the numerator by $x^{15}$: $2x^{12}+5x^9 = \frac{2x^{12}}{x^{15}} + \frac{5x^9}{x^{15}} = 2x^{-3} + 5x^{-6}$

Divide the denominator by $x^{15}$: $(x^5+x^3+1)^3 = \left(\frac{x^5+x^3+1}{x^5}\right)^3 = \left(1+\frac{x^3}{x^5}+\frac{1}{x^5}\right)^3 = (1+x^{-2}+x^{-5})^3$

So, the integral becomes: $I = \int \frac{2x^{-3}+5x^{-6}}{(1+x^{-2}+x^{-5})^3} dx$

Step 2: Perform Substitution

Let $u = 1+x^{-2}+x^{-5}$. Now, find the differential $du$: $du = \frac{d}{dx}(1+x^{-2}+x^{-5}) dx$ $du = (0 + (-2)x^{-2-1} + (-5)x^{-5-1}) dx$ $du = (-2x^{-3} - 5x^{-6}) dx$ $du = -(2x^{-3} + 5x^{-6}) dx$

From this, we can see that $(2x^{-3} + 5x^{-6}) dx = -du$.

Step 3: Substitute into the integral and evaluate

Substitute $u$ and $du$ into the modified integral: $I = \int \frac{-du}{u^3}$ $I = -\int u^{-3} du$

Now, integrate using the power rule $\int y^n dy = \frac{y^{n+1}}{n+1} + C$: $I = - \left( \frac{u^{-3+1}}{-3+1} \right) + C$ $I = - \left( \frac{u^{-2}}{-2} \right) + C$ $I = \frac{1}{2u^2} + C$

Step 4: Substitute back for x

Replace $u$ with $1+x^{-2}+x^{-5}$: $I = \frac{1}{2(1+x^{-2}+x^{-5})^2} + C$

To match the options, convert $1+x^{-2}+x^{-5}$ back to a common denominator form: $1+x^{-2}+x^{-5} = 1 + \frac{1}{x^2} + \frac{1}{x^5} = \frac{x^5}{x^5} + \frac{x^3}{x^5} + \frac{1}{x^5} = \frac{x^5+x^3+1}{x^5}$

Substitute this back into the expression for $I$: $I = \frac{1}{2 \left( \frac{x^5+x^3+1}{x^5} \right)^2} + C$ $I = \frac{1}{2 \frac{(x^5+x^3+1)^2}{(x^5)^2}} + C$ $I = \frac{1}{2 \frac{(x^5+x^3+1)^2}{x^{10}}} + C$ $I = \frac{x^{10}}{2(x^5+x^3+1)^2} + C$

Comparing this result with the given options, it matches option (3).