Grade 12 · Math · Calculus

12th grade Integrals with inverse trig, long division and completing the square worksheet

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Integrals with inverse trig, long division and completing the square

Grade 12 Math Worksheet

NameDate

Circle the best answer, or write your answer on the line.

Ari evaluates ∫ from 0 to 2 of 8/(4 + x²) dx with the arctan rule. What is it, to the nearest hundredth?
Bo evaluates ∫ from 0 to 2 of 6/(4 + x²) dx with the arctan rule. What is it, to the nearest hundredth?
Esme evaluates ∫ from 0 to 5 of 10/(25 + x²) dx with the arctan rule. What is it, to the nearest hundredth?
Jun evaluates ∫ from 0 to 3 of 15/(9 + x²) dx with the arctan rule. What is it, to the nearest hundredth?
5.
Kira divides first and finds ∫ (x² + 3x + 5)/(x + 1) dx. Which is it?
- x²/2 + 2x + 3 ln|x + 1| + C
- x²/2 + 2x + C
- (x³/3 + 3x²/2 + 5x)/(x²/2 + x) + C
6.
Luz divides first and finds ∫ (x² + 4)/(x − 2) dx. Which is it?
- x²/2 + 2x + C
- (x³/3 + 4x)/(x²/2 − 2x) + C
- x²/2 + 2x + 8 ln|x − 2| + C
7.
Milo divides first and finds ∫ (2x² + x)/(x + 1) dx. Which is it?
- x² − x + ln|x + 1| + C
- x² − x + C
- (2x³/3 + x²/2)/(x²/2 + x) + C
Pia completes the square to evaluate ∫ from 1 to 3 of 1/(x² − 2x + 5) dx. What is it, to the nearest hundredth?
Ren completes the square to evaluate ∫ from −2 to 1 of 1/(x² + 4x + 13) dx. What is it, to the nearest hundredth?
Zee completes the square to evaluate ∫ from 5 to 10 of 1/(x² − 10x + 50) dx. What is it, to the nearest hundredth?

Integrals with inverse trig, long division and completing the square. Every version has 10 questions, and the answer key prints on the second page.

Standard: none (beyond CCSS)

arctan rule: ∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C.

long division: Split a top-heavy fraction into a polynomial and a remainder fraction.

complete the square: Rewrite x² + bx + c as (x + h)² + k.

Used the wrong inverse trig rule: 1/(1 + x²) goes with arctan and 1/√(1 − x²) with arcsin.

Used ln of the bottom, which only works when the top is the bottom's derivative.

Left out the 1/a in front: ∫ 1/(a² + x²) dx = (1/a) arctan(x/a).

What is the most common mistake with integrals with inverse trig, long division and completing the square?

Used the wrong inverse trig rule: 1/(1 + x²) goes with arctan and 1/√(1 − x²) with arcsin.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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