find the indefinite integral

Question

16x(8x^2 + 9)^4 dx

Answer 1

Inspecting 16x(8x² + 9)⁴, one realizes that 16x is the derivative of 8x² + 9.

To build the integral, first increase the power of (8x² + 9) from 4 to 5.

Then supply (1)(8x² + 9)⁵ with a coefficient that will multiply by the power exponent 5 to produce 1.

Stand (1/5) or 0.2 by (8x² + 9)⁵ to obtain 0.2(8x² + 9)⁵.

Make a test differentiation of 0.2(8x² + 9)⁵ which goes to (0.2)(5)(16x + 0)(8x² + 9)^(5-1).

Simplify (0.2)(5)(16x + 0)(8x² + 9)^(5-1) to 16x(8x² + 9)⁴, the integrand given in the problem statement.

The Indefinite Integral is then written as ∫16x(8x² + 9)⁴dx = 0.2(8x² + 9)⁵ + C (where C represents some exact number, as opposed to a variable number, that will vanish to 0 when differentiated).