Need Simple Calculus Help! Antiderivatives, ln and e Help!?

Question

I need to find the antiderivative of f(x) = e ^ (-3x) Step by step help in solving this would be greatly appreciated. Thanks so much! ALSO:
What if I need to find the antiderivative of a constant, say f(x) = 3?

Answer 1

antiderivative = integration

first we know, differentiation of e^(mx) = m e^(mx)

so integration of m.e^(mx) = e^(mx) (of course +c )
or, integration of e^(mx) = 1/m e^(mx) + c

hence integration of e^(-3x) = -1/3 e^(-3x) + c

step by step: assume -3x=y then - 3dx=dy hence the given problem is equivalent to integrating -1/3 e^y w.r.t. dy

differentiation of x w.r.t x is 1
differentiation of constant w.r.t.anything = 0

so integration of 3 wrt x = 3x+c (constant is arbitrary)

important: read calculus basics again & again !

Answer 2

Antiderivatives = Undoing differentiation

A function of F is called an antiderivative of f on an interval I if F' (x) = f(x) for all functions.

If F is an antiderivative of f in any interval I then the most general antiderivative of f on I is F(x)+C where C is any constant.

Answer 3

Use a u substitution.
let u = -3x
du = -3 dx

You may need to study your derivitive rules. There usually is no set method for finding the antiderivitive. You just need to remember your derivitive rules and recognize when you have a derivitive similar to one.

For your first problem, remember that the derivitive of e^x is e^x.

for f(x) = 3, you need to remember some function that would give you the derivitive of 3. since a derivitive gives the slope of the equation at a particular point. f(x) = 3 tells you that the slope is 3 everywhere. This only happens when you have a line of slope 3. y = 3x. You can also rewrite f(x) = 3 as f(x) = 3 * x^0 and use the power rule.

Answer 4

In general, the antiderivative of k is kx + C where C is any constant. In this case (k = 3, C = 0), the derivative of 3x is 3.

Given that the derivative of e ^ x is itself and by using chain rule, you can determine that the derivative of e ^ (nx) is ne ^ (nx). Since dividing both the initial function and its derivative by a constant still gives a correct result, you can divide by n and determine that the derivative of e ^ (nx) / n is e ^ (nx).

Reverse the order and you know that the antiderivative of e ^ (nx) is e ^ (nx) / n.

For your specific example (n = -3) the antiderivative is (-1/3) e ^ (-3x).

Answer 5

rfamilymember shows the first correct answer. Always try substitution first or integration by parts. Be aware that sometimes you cannot find an antiderivative and are thus forced to use numerical methods.

Answer 6

The antiderivative of f(x) = e ^ (-3x) is the indefinite integral of f(x) with respect to x. The same is true with f(x) = 3.

Answer 7

Antiderivative of a constant is usually the constant multplied by a variable.

Answer 8

let -3x=t
-3dx=dt
dx=-dt/3
antiderivative of (-1/3)e^tdt=(-1/3)e^t+C
=(-1/3)e^(-3x)+c

Answer 9

Dude, can't you read your textbook?

Answer 10

wow...u r smart...
i am also smart...