Integration Techniques

Strategies for indefinite u-sub

• make u equal to expression that’s more complicated

• set u to denominator (especially for trig functions)

• make u equal to expression with higher exponent

• make it equal to whatever will cancel the other expression in the integral with X

Can you choose to not make an expression u even if u equals that expression?

Yes (do this when you need to cancel x and u doesn’t help)

What to do when x is still in the integral after u subbing

• manipulate u equation to equal x in terms of u and plug into it into integral

Trig u-sub strategies

• use identities to isolate a single trig function with a power of 1, and create u so it will cancel (usually u is the function you have more of so the single expression will cancel)

• use identities to convert between sines and cosines

• make u the trig function that you have a higher power of usually

• sin⁴x = (sin²x)² works for cosine too

Is the domain of a function that same as its integral?

Yes

When would you add absolute value bars when solving the integral?

• When the antiderivative has a log function where negatives can’t be inputted, and the domain of the original function is all real numbers

When would you NOT add absolute value bars inside a log function of an antiderivative?

when the function inside the log always evaluates to positive no matter x (i.e. e^x)

How to solve definite integrals using u-sub

• solve exact same way using u sub, except use 1st Fund Theorem at the end to solve the antiderivative at the bounds

How to change the bounds of a definite integral

• Find u

• Plug in original top and bottom bounds for x in the u equation and set new values for top and bottom respectively

• If new top value turns out to be less than the new bottom value, flip the integral

• in antiderivative don’t plug x back, keep u and sub in for 1st FTC

If you change the bounds of a definite integral, do you have to plug x back in for u?

No (just plug in new top and bottom bounds for u using 1st FTC)

Differential equation

equation that involves a derivative

What does a slope field show?

the slope of the antiderivative at each point on a graph

Strategy for drawing slope fields for differential equations with x and y variables

• Pick one x coordinate and plug it in with all other y coordinates in the graph to find the slopes at every y for that x

How to draw the graph of the antiderivative on a slope field from a given point

• plot the point and then follow the curves on each slope line from there

Strategies for matching a slope field to a differential equation

• Look for horizontal tangents at corners

• test x = 0

• pick points to test

If a trig integral already has sine and cosine, and one of them has a power of 1 while the other has a higher power, make u….

equal to the function in the higher power

(if integral is sin(x)cos³(x), make u = cos(x)

If an initial condition is given, use it find…

C

What is the integral of dx?

x

How to complete the square

1. Move c in quadratic out of the expression, to make room for a new c

2. (b/2)² = new c value in quadratic

3. Condense the quadratic into perfect square

4. Subtract the new c from the old c

If there’s a negative outside of the quadratic when completing the square, then ____ the new c value to the old c

add

If there’s a negative in front of x² and you need to complete the square, then…

factor out the negative only for a and b in the quadratic

For synthetic division, the divisor has to be in the form of…

x - c

(if x + c, make c negative in the division)

If you don’t have 1 for the constant in the denominator when completing the square, then…

• multiply the numerator and denominator by the reciprocal of the constant

• Incorporate this value into the parentheses of the perfect square by taking it’s square root and placing it in the denominator of the expression

For undefined slopes in a slope field, draw…

no lines

If you get a plus or minus when integrating by separation of variables, how do you determine which one to use?

• Use initial condition and plug into the positive and negative equation

• See which one actually yields a true answer

What to do if you integrate by separating variables and you get an equation that doesn’t make the initial condition true

• If you use an absolute value sign when integrating, get rid of it and make the equation on the other side of the equal sign plus or minus

• Take the sign that makes the initial condition true