Find the indefinite integral

Question

/8xdx and /x^5dx

Answer 1

Remember that, in a certain sense, we can think of integration as the inverse of differentiation. In the same way that the division question 6 ÷ 2 asks "the product of what number and 2 equals 6?" an integral like ∫8x dx asks "what function differentiated gives us the function 8x?".

You might remember a differentiation rule called the "power rule," which tells us how to deal with functions of a variable raise to a power. The derivative dy/dx of y = x², for instance, is 2x, and in general, differentating the function y = xⁿ with respect to n gives us the derivate dy/dx = nx^n-1. We multiply the variable by its power and decrease its power by one. We can also follow this process in reverse: to find the function we need to differentiate to get 2x, we know that function's power must be one higher - which in this case is 1 + 1 = 2. We can divide 2x by 2 and increase its power to get the function x², restoring our original function.

One more important detail: we have to keep in mind that we lose information while differentiating. Since derivatives are all about the rate of change of functions, we disregard any constants added to the end of a function. y = x², y = x² + 3, and y = x² - 42 all have the same derivative: dy/dx = 2x. When we integrate the function 2x, we account for this ambiguity by adding a constant to the end, so ∫2x dx = x² + c. Sometimes you'll see this called the constant of integration.

Taken together with the "reverse power rule", the integral of a power function ∫axⁿ dx = (axⁿ⁺¹)/n + c.

Applying this rule to our expressions gives us:

∫8x dx = (8x²)/2 + c = 4x² + c

∫x⁵ dx = x⁶/6 + c

Answer 2

4x^2 + c

nad

(1/6)x^6 + c