Calculus (MATH 1500): Antiderivatives

Give the antiderivatives of the following: ∫ 0 dx, ∫ 1 dx, ∫ xⁿ dx, ∫ e^x dx.

• ∫ 0 dx = 1 + C

• ∫ 1 dx = x + C

• ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (n ≠ -1)

• ∫ e^x dx = e^x + C

Give the antiderivatives of the following: ∫ e^kx dx, ∫ b^x dx, ∫ 1/x dx, ∫ cos(x) dx.

• ∫ e^kx dx = e^kx/k + C

• ∫ b^x dx = b^x/ln(b) + C

• ∫ 1/x dx = ln|x| + C (only when x^-1!)

• ∫ cos(x) dx = sin(x) + C

Give the antiderivatives of the following: ∫ sin(x) dx, ∫ sec²(x) dx, ∫ csc²(x) dx, ∫ sec(x)tan(x) dx.

• ∫ sin(x) dx = -cos(x) + C

• ∫ sec²(x) dx = tan(x) + C

• ∫ csc²(x) dx = -cot(x) + C

• ∫ sec(x)tan(x) dx = sec(x) + C

Give the antiderivatives of the following: ∫ csc(x)cot(x) dx, ∫ 1/√(1-x²) dx, ∫ 1/(1+x²) dx.

• ∫ csc(x)cot(x) dx = -csc(x) + C

• ∫ 1/√(a²-x²) dx = arcsin(x/a) + C

• ∫ 1/(a²+x²) dx = 1/a arctan(x/a) + C

Give the antiderivatives of the following:

• ∫ tanx dx = -ln|cosx|

• ∫ cotx dx = ln|sinx|

• ∫ secx dx = ln|secx+tanx|

• ∫ csc dx = -ln|cscx+cotx|

When should I think “logarithm” for an antiderivative?

If the numerator is (or is a constant multiple of) the derivative of the denominator:

• ∫ f’(x)/f(x) dx = ln|f(x)| + C

How do find a function such that you have its given implicit derivative, input, and output are given?

> Multiply by dx on both sides and ensure x and y variables are separated.

> Integrate both sides.

> Sub the input x and output y → solve for C.

> Rewrite antiderivative with the solved C.

How do you solve particle motion problems given force, acceleration, velocity, or position?

> Convert units if needed.

> Convert physics into calculus (F = ma) → Find acceleration a(t).

> Integrate to get velocity v(t) → Use initial velocity to find C.

> Integrate to get position s(t) → Use initial position to find C.

> To find time at max height, etc., set v(t) = 0.

> Plug that time into s(t).

What does FTC 1 state?

If

$$g\left(x_{}\right)=\int_{a}^{x}\!f\left(t\right)\,dt\Rightarrow g^{\prime}\left(x\right)=f\left(x\right)$$

or rather

\frac{d}{\differentialD x}\left\lbrack\int_{a}^{x}f\left(t\right)\!\,dt\right\rbrack=f\left(x\right)

The derivative of an integral is the original function.

How do you show proof of FTC 1?

> Convert to d/dx [ ∫_a^x f(t) dt].

→ d/dx [F(x)|_a^x]

→ d/dx [F(x) - F(a)] = d/dx F(x) - d/dx F(a) = F’(x)

> Solve.

What does FTC 2 state?

If

$$F^{\prime}\left(x\right)=f\left(x\right)$$

then

$$\int_{a}^{b}\!f\left(x\right)\,dx=F\left(b\right)-F\left(a\right)$$

A definite integral equals the antiderivative evaluated at the endpoints.

How do you show proof of FTC 2?

> Get the antiderivative F(x).

> Solve for F(b) - F(a).

How do you compute the Riemann sum for the partition of a given interval with n equal-sized subintervals with the left-, right-, and mid-point approximation?

> Find ∆x = b - a / n.

> Create partition points and number line.

> Choose sample points.

• Left: x₀, x₁, … x_n-1

• Right: x₁, x₂ … x_n

• Midpoint: midpoints of each subinterval.

> Compute A = ∆x ∑ f(x_i).

> IF NEEDED: Compute definite integral and compare if under- or over-estimation.

is L_n always an underestimate and R_n always an overestimate?

> For an increasing function, left is an under- and right is an over-estimate.

> For a decreasing function, right is an under- and left is an over-estimate.

You can check f’(x) > 0 if increasing or f’(x) < 0 for decreasing.

How do you compute the area under a given graph with its equation on a given interval?

Use FTC 2.

How do you compute the total area of a graph over an interval?

> Find where the graph crosses the x-axis by setting the f(x) = 0 and solving for x (which we will now call b).

> Split at b, where a = lower limit and c = upper limit: (-1) ∫_a^b f(x) dx + ∫_c^b f(x) dx.

> Find the antiderivative → Solve each integral and ensure the ∫_a^b is positive due to (-1).

> Add the positive areas → A_tot.

How do you show that f(x) ≤ ∫_a^x g(x) dx ≤ h(x)

> Look at the interval. Find its length by adding the endpoints together.

> Find the smallest value (f) and largest value (h) by evaluating f(x) with x as the lower and upper limits.

> f ≤ f(x) ≤ h

> Integrate all three parts.

Formula for mean value of f(x) on interval