Calculate the integral of (7x^5) dx.

Calculate the integral of (7x^5) dx.

• A. (7/6)x^6 + C
• B. (7/5)x^6 + C
• C. (6/7)x^6 + C
• D. 7x^6 + C

Answer: (A)

To calculate the integral of \(7x^5\) with respect to \(x\), we use the power rule for integration. The power rule states that the integral of \(x^n\) is given by \(\frac{x^{n+1}}{n+1}\), plus a constant of integration \(C\), where \(n\) is any real number except \(-1\). In this case, we have the function \(7x^5\). We can factor out the constant (7) before applying the power rule: \[ \int 7x^5 \, dx = 7 \int x^5 \, dx \] Now applying the power rule, we increase the exponent by 1 (from 5 to 6) and divide by the new exponent (6): \[ \int x^5 \, dx = \frac{x^{6}}{6} \] So now substituting back, we get: \[ 7 \int x^5 \, dx = 7 \cdot \frac{x^{6}}{6} = \frac{7}{6}x^{6} \] Finally, we add the constant of integration \(C

To calculate the integral of (7x^5) with respect to (x), we use the power rule for integration. The power rule states that the integral of (x^n) is given by (\frac{x^{n+1}}{n+1}), plus a constant of integration (C), where (n) is any real number except (-1).

In this case, we have the function (7x^5). We can factor out the constant (7) before applying the power rule:

[

\int 7x^5 , dx = 7 \int x^5 , dx

]

Now applying the power rule, we increase the exponent by 1 (from 5 to 6) and divide by the new exponent (6):

[

\int x^5 , dx = \frac{x^{6}}{6}

]

So now substituting back, we get:

[

7 \int x^5 , dx = 7 \cdot \frac{x^{6}}{6} = \frac{7}{6}x^{6}

]

Finally, we add the constant of integration (C