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 - Building Confidence

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

1.
Bo finds ∫ 3/√(1 − x²) dx using an inverse trig rule. Which is it?
- 3 arctan x + C
- 3 ln(1 − x²) + C
- 3 arcsin x + C
2.
Esme finds ∫ 2/(9 + x²) dx using an inverse trig rule. Which is it?
- (2/3) arcsin(x/3) + C
- (2/3) arctan(x/3) + C
- 2 arctan(x/3) + C
3.
Jun finds ∫ 4/(16 + x²) dx using an inverse trig rule. Which is it?
- arctan(x/4) + C
- 4 arctan(x/4) + C
- arcsin(x/4) + C
Kira divides first: (x² + 3)/x = x + 3/x. What is ∫ from 1 to 2 of (x² + 3)/x dx, to the nearest hundredth?
Luz divides first: (x² + 5)/x = x + 5/x. What is ∫ from 1 to 3 of (x² + 5)/x dx, to the nearest hundredth?
Milo divides first: (x² + 2)/x = x + 2/x. What is ∫ from 1 to 4 of (x² + 2)/x dx, to the nearest hundredth?
Noah divides first: (x² + 6)/x = x + 6/x. What is ∫ from 1 to 2 of (x² + 6)/x dx, to the nearest hundredth?
8.
Pia completes the square in x² + 2x + 5 to find ∫ 1/(x² + 2x + 5) dx. Which is it?
- arctan((x + 1)/2) + C
- (1/2) arctan((x + 1)/2) + C
- (1/4) arctan((x + 1)/2) + C
- (1/2) arctan((x − 1)/2) + C
9.
Tala completes the square in x² + 6x + 25 to find ∫ 1/(x² + 6x + 25) dx. Which is it?
- (1/4) arctan((x + 3)/4) + C
- (1/16) arctan((x + 3)/4) + C
- arctan((x + 3)/4) + C
- (1/4) arctan((x − 3)/4) + C
10.
Zee completes the square in x² − 8x + 41 to find ∫ 1/(x² − 8x + 41) dx. Which is it?
- (1/5) arctan((x + 4)/5) + C
- (1/5) arctan((x − 4)/5) + C
- arctan((x − 4)/5) + C
- (1/25) arctan((x − 4)/5) + C

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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