Evaluate the integral.

Question

BlackTom AI · Accepted Answer

We are asked to evaluate the integral ∫ sin(9x - 3) dx.

First, recall the standard antiderivative rule: ∫ sin(kx + c) dx = -cos(kx + c)/k + C, for any nonzero constant k. This comes from the chain rule in reverse, where differentiating cos(kx + c) yields -k sin(kx + c).

Applying the rule with k = 9 and c = -3, we obtain: ∫ sin(9x - 3) dx = -cos(9x - 3) / 9 + C.

Now, evaluate the given answer choices conceptually:
- If an option presents -cos(9x - 3) + C without the division by 9, it is missing the factor 1/9. Differentiating -cos(9x - 3) would give 9 sin(9x - 3), not sin(9x - 3), so this option does not match the original integrand upon differentiation.

- If an option includes - (1/9) cos(9x - 3) + C, that aligns with the correct antiderivative, since differentiating -(1/9) cos(9x - 3) yields sin(9x - 3) by the chain rule.

- If an option shows a positive cos(9x - 3) term, i.e., +cos(9x - 3) or +(1/9) cos(9x - 3), these would differentiate to -sin(9x - 3) (or -(1/9) sin(9x - 3) after the chain rule), which does not match sin(9x - 3). Thus they are incorrect.

- Any option that omits the constant of integration C is incomplete, since indefinite integrals include an arbitrary constant.

In summary, the correct antiderivative must have a factor of -1/9 in front of cos(9x - 3) and include + C to satisfy the indefinite integral form. The other forms fail either by missing the 1/9 factor, changing the sign incorrectly, or omitting the constant of integration.

MATH2415W16 Quiz 4 Dec 16 (W16)- Requires Respondus LockDown Browser

Evaluate the integral.

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MATH*2415*W16 Quiz 4 Dec 16 (W16)- Requires Respondus LockDown Browser

Evaluate the integral.

View Explanation

Log in for full answers

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Find the integral ∫ 𝑥 2 ( 2 𝑥 3 + 1 ) 4 𝑑 𝑥

∫ sec 2 ⁡ 𝑥 tan 7 ⁡ 𝑥 𝑑 𝑥 = _ _ _ _ _ _ _ _ _ _ _

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Question at position 5 ∫323dx=\int3^{\frac{2}{3}}dx=323x+C3^{\frac{2}{3}}x+C35353+C\frac{3}{5}3^{\frac{5}{3}}+C53353+C\frac{5}{3}3^{\frac{5}{3}}+C000+C0+C

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MATH2415W16 Quiz 4 Dec 16 (W16)- Requires Respondus LockDown Browser