Question

int (7x^{6} + 7x^{3} + 4) dx = \boxed{} + c

Explanation:

Step1: Integrate \(7x^6\)

Using the power rule \(\int x^n dx = \frac{x^{n + 1}}{n+1} + C\) (for \(n
eq - 1\)), for \(n = 6\), we have \(\int7x^{6}dx=7\times\frac{x^{6 + 1}}{6+1}=7\times\frac{x^{7}}{7}=x^{7}\)

Step2: Integrate \(7x^3\)

For \(n = 3\), \(\int7x^{3}dx=7\times\frac{x^{3+1}}{3 + 1}=7\times\frac{x^{4}}{4}=\frac{7x^{4}}{4}\)

Step3: Integrate \(4\)

For the constant function \(4\) (which can be written as \(4x^{0}\)), using the power rule with \(n = 0\), \(\int4dx=4\times\frac{x^{0 + 1}}{0+1}=4x\)

Step4: Combine the results

Adding the results of the three integrations together, we get \(x^{7}+\frac{7x^{4}}{4}+4x\)

Answer:

\(x^{7}+\frac{7x^{4}}{4}+4x\)