Integral Calculus Practice test

What is an antiderivative?

A function F whose derivative is f (F' = f)

What is the general form of an indefinite integral?

F(x) + C

What is the integral of k with respect to x?

kx + C

Compute ∫ x² dx.

x³/3 + C

Compute ∫ (3x² − 4x + 1) dx.

x³ − 2x² + x + C

What is the derivative of an integral?

By FTC: d/dx ∫_a^x f(t) dt = f(x)

State the Fundamental Theorem of Calculus (Part 1).

∫_a^b f'(x) dx = f(b) − f(a)

State the Fundamental Theorem of Calculus (Part 2).

If F' = f, then ∫_a^b f(x) dx = F(b) − F(a)

Evaluate ∫₀³ (2x) dx.

9

Evaluate ∫₁⁴ (1/x) dx.

ln(4)

Compute ∫ (1/(x ln x)) dx.

ln|ln x| + C

Find ∫ e^(3x) dx.

(1/3)e^(3x) + C

Find ∫ cos(5x) dx.

(1/5) sin(5x) + C

Find ∫ sec²(2x) dx.

(1/2) tan(2x) + C

Compute ∫ (1/(1+x²)) dx.

arctan(x) + C

Evaluate ∫ (1/√(1−x²)) dx.

arcsin(x) + C

Evaluate ∫ 2x(3x² + 1)^5 dx.

(3x² + 1)^6/3 + C

What substitution is used for √(a² − x²)?

x = a sin θ

What substitution is used for √(a² + x²)?

x = a tan θ

What substitution is used for √(x² − a²)?

x = a sec θ

Compute ∫₀¹ (x³ + 2) dx.

(1/4) + 2

Conceptually, what does a definite integral represent?

Net area between curve and x-axis

When is the area between curve and x-axis negative?

When f(x) < 0

Compute the area under y = 4 − x² from x = 0 to x = 2.

16/3

Find the area between y = x and y = x² on [0,1].

1/6

What is the disk method formula for volume?

V = π ∫ (R(x))² dx

What is the washer method formula for volume?

V = π ∫ [R(x)² − r(x)²] dx

What is the shell method formula for volume?

V = 2π ∫ (radius)(height) dx

Find volume of a solid formed by rotating y = x² around the x-axis on [0,2].

8π/5

Find volume using shells for region under y = x rotated around y-axis on [0,3].

27π/2

Compute ∫ (1/x³) dx.

-1/(2x²) + C

Compute ∫ tan x dx.

−ln|cos x| + C

Compute ∫ cot x dx.

ln|sin x| + C

Compute ∫ sec x dx.

ln|sec x + tan x| + C

Compute ∫ csc x dx.

ln|csc x − cot x| + C

Compute ∫ ln x dx.

x ln x − x + C

Compute ∫ x e^x dx.

x e^x − e^x + C

Compute ∫ x cos x dx.

x sin x + cos x + C

Compute ∫ e^x sin x dx.

(1/2)e^x(sin x − cos x) + C

Evaluate ∫₁^e (ln x) dx.

e − 1

Evaluate ∫₀^∞ e^(−x) dx.

1

Determine if ∫₁^∞ (1/x²) dx converges.

Yes (equals 1)

Determine if ∫₁^∞ (1/x) dx converges.

No (diverges)

Compute ∫₀^∞ e^(−2x) dx.

1/2

Compute ∫₀^4 |2x − 4| dx.

8

Find ∫ (3x − 4)(x² − 1) dx.

(3/4)x⁴ − 2x³ − (3/2)x² + 4x + C

Differentiate ∫₀^x (t² + 1) dt.

x² + 1

Find ∫ (1/(√x + 1)) dx.

2√x − ln|√x − 1| + C

Find ∫ (x / √(4 − x²)) dx.

−√(4 − x²) + C

Find ∫ (x² / (1 + x³)) dx.

(1/3) ln|1 + x³| + C

Compute d/dx [ ∫₁^{x²} ln t dt ].

2x ln(x²)

Find ∫ e^(−x²) dx.

No elementary antiderivative (Error function)

Compute ∫ 4/(x² + 4) dx.

2 arctan(x/2) + C

Compute ∫ 6x/(x² + 9) dx.

3 ln(x² + 9) + C

Compute ∫ (5/(√(25 − x²))) dx.

5 arcsin(x/5) + C

Find ∫ sin³x dx.

−cos x + (1/3)cos³x + C

Find ∫ cos³x dx.

sin x − (1/3) sin³x + C

Find ∫ sec³x dx.

(1/2)(sec x tan x + ln|sec x + tan x|) + C

Compute ∫ x√(x+1) dx.

(2/15)(x+1)^(5/2)(3x−2) + C

Find ∫ (1/(x² − 4)) dx.

(1/4) ln|(x−2)/(x+2)| + C

Evaluate ∫₀^1 (1/(1+x³)) dx.

No elementary antiderivative (leave as integral)

Find ∫ dx/(x(ln x)²).

−1/ln x + C

Find ∫ e^(2x) sin(3x) dx.

(e^(2x)(2 sin 3x − 3 cos 3x))/13 + C

Find the integral that represents the arclength of y = x² on [0,2].

∫₀² √(1 + (2x)²) dx

Find ∫ (d/dx)(x³ e^{2x}) dx.

x³ e^{2x} + C

Compute ∫ (1/(x³ + 1)) dx.

Partial fractions required

Conceptually, what is an improper integral?

An integral where limits or integrand are infinite

Determine if ∫₀^∞ (1/(1 + x²)) dx converges.

Yes (equals π/2)

Determine if ∫₀^∞ x e^(−x) dx converges.

Yes (equals 1)