Evaluate the integral. (Use c for the constant of integration.) ∫9⁢sin2⁢(x)cos3⁢(x) dx

9 ⁢ ( 1 5 sin 3 ⁢ ( x ) ) - 9 ⁢ ( 1 7 sin 7 ⁢ ( x ) ) + c

Evaluate the integral. (Use c for the constant of integration.) ∫ 9 ⁢ sin 2 ⁢ ( x ) cos 3 ⁢ ( x ) d x

9 ⁢ ( 1 5 sin 3 ⁢ ( x ) ) - 9 ⁢ ( 1 7 sin 7 ⁢ ( x ) ) + c

9 ⁢ ( 1 5 cos 5 ⁢ ( x ) ) + c

9 ⁢ ( 1 5 sin 5 ⁢ ( x ) ) + c

9 ⁢ ( 1 3 sin 3 ⁢ ( x ) cos 3 ⁢ ( x ) ) + c

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7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

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Multiple Choice

Evaluate the integral. (Use c for the constant of integration.) $\int 9 \sin^{2} (x) \cos^{3} (x) d x$

9 (\frac{1}{5} \cos^{5} (x)) + c

9 (\frac{1}{5} \sin^{5} (x)) + c

9 (\frac{1}{3} \sin^{3} (x) \cos^{3} (x)) + c

9 (\frac{1}{5} \sin^{3} (x)) - 9 (\frac{1}{7} \sin^{7} (x)) + c

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Verified step by step guidance

Step 1: Recognize that the integral involves trigonometric functions raised to powers. To simplify, use trigonometric identities or substitution methods. Specifically, consider using the identity $ \sin^2(x) = 1 - \cos^2(x) $ to rewrite $ \sin^2(x) $ in terms of $ \cos(x) $.

Step 2: Rewrite the integral $ \int 9 \sin^2(x) \cos^3(x) \, dx $ as $ \int 9 (1 - \cos^2(x)) \cos^3(x) \, dx $. This simplifies the expression and prepares it for integration.

Step 3: Expand the integrand to separate terms: $ \int 9 \cos^3(x) \, dx - \int 9 \cos^5(x) \, dx $. Now, you have two simpler integrals to evaluate.

Step 4: Use substitution for each term. Let $ u = \cos(x) $, then $ du = -\sin(x) \, dx $. Substitute into the integral, transforming it into a polynomial in $ u $. For example, $ \int \cos^3(x) \, dx $ becomes $ \int u^3 (-du) $.

Step 5: Integrate each term in $ u $-form. For $ \int u^3 (-du) $, the result is $ -\frac{u^4}{4} $. Similarly, integrate $ \int u^5 (-du) $ to get $ -\frac{u^6}{6} $. Substitute back $ u = \cos(x) $ and combine terms to express the final result in terms of $ \sin(x) $ and $ \cos(x) $. Add the constant of integration $ c $.