Question

find \\(\int\left( \dfrac{7}{x^6} + 6x + 4 \
ight) dx\\) \\(\boxed{}\\) + c

Explanation:

Step1: Rewrite the rational term

Rewrite $\frac{7}{x^6}$ as $7x^{-6}$

Step2: Integrate term by term

Apply power rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for each term:

• For $7x^{-6}$: $\int 7x^{-6} dx = 7 \cdot \frac{x^{-6+1}}{-6+1} = 7 \cdot \frac{x^{-5}}{-5} = -\frac{7}{5x^5}$
• For $6x$: $\int 6x dx = 6 \cdot \frac{x^{1+1}}{1+1} = 6 \cdot \frac{x^2}{2} = 3x^2$
• For $4$: $\int 4 dx = 4x$

Step3: Combine all integrated terms

Sum the results of each integration.

Answer:

$-\frac{7}{5x^5} + 3x^2 + 4x$