Derivatives and Integrals

Calculus Tutorial

Derivatives and Integrals

Intro

Fine a derivative and an integral that are the same.

Sample Problem

a) fine the derivative. Cos(2x)-6x²

b) fine the integral. ∫ cos(2x)-6x² dx

Solution

a) cos(2x)-6x²

We con solve cos(2x) apart from 6x² because of the – .

dx of cos(2x) = -sin(2x) dx 2x
dx of 2x = 2
dx of cos(2x) = -2sin(2x)

dx of 6x²=12z

Together the answer is -2sin(2x)-12x

b) ∫ cos(2x)-6x² dx

Here we also, can solve cos(2x) and 6x² apart, because of the – .

∫ cos(2x) dx

For this integral we need to use U sub.
U = 2x du = 2 dx du/2 = dx
du/2 = 1/2 that goes out side the integral.
1/2 ∫ cos(U) du
∫ cos(U) = sin(U)
Put 2x back in for U
∫ cos(2x) = 1/2(sin(2x)

∫ 6z² dx = 6x³/3 or if simplified 2x³

answer 1/2 sin(2x)-2x³+c



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