Find \( \int (3x^{3} + 4x^{5} - 7) \, dx - Edexcel - A-Level Maths Pure - Question 3 - 2008 - Paper 2

To find the integral of the polynomial, we apply the power rule of integration which states that ( \int x^{n} , dx = \frac{x^{n+1}}{n+1} + C ), where ( C ) is the constant of integration.

1. Integrating the first term: [ \int 3x^{3} , dx = 3 \cdot \frac{x^{3+1}}{3+1} = \frac{3x^{4}}{4} ]

2. Integrating the second term: [ \int 4x^{5} , dx = 4 \cdot \frac{x^{5+1}}{5+1} = \frac{4x^{6}}{6} ]

3. Integrating the third term: [ \int -7 , dx = -7x ]

Putting it all together, we find: [ \int (3x^{3} + 4x^{5} - 7) , dx = \frac{3x^{4}}{4} + \frac{4x^{6}}{6} - 7x + C ]

This can also be expressed in a simplified form: [ \frac{3x^{4}}{4} + \frac{2x^{6}}{3} - 7x + C ]

Thus, the final answer is: [ \int (3x^{3} + 4x^{5} - 7) , dx = \frac{3x^{4}}{4} + \frac{2x^{6}}{3} - 7x + C ]