Evaluate the integral. int(-16x^(3)-21x^(2))dx Answer:

Apply Power Rule: Apply the power rule for integration to each term separately. The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1) + C, where C is the constant of integration.Integrate -16x^3: Integrate the first term -16x^3. Using the power rule, we add 1 to the exponent and divide by the new exponent. Integral of -16x^3 dx = -16 * (x^(3+1))/(3+1) = -16 * (x^4)/4 = -4x^4Integrate -21x^2: Integrate the second term -21x^2. Using the power rule, we add 1 to the exponent and divide by the new exponent. Integral of -21x^2 dx = -21 * (x^(2+1))/(2+1) = -21 * (x^3)/3 = -7x^3Combine Results: Combine the results of the integrals of both terms and add the constant of integration C. Integral of (-16x^3 - 21x^2) dx = -4x^4 - 7x^3 + C

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Find indefinite integrals using the substitution

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Q. Evaluate the integral.

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\int\left(-16 x^{3}-21 x^{2}\right) d x

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Answer:

Apply Power Rule: Apply the power rule for integration to each term separately. The power rule states that the integral of $x^n$ with respect to $x$ is $(x^{(n+1)})/(n+1) + C$ , where $C$ is the constant of integration.
Integrate $-16x^3$ : Integrate the first term $-16x^3$ . Using the power rule, we add $1$ to the exponent and divide by the new exponent. Integral of $-16x^3 \, dx = -16 \times (x^{3+1})/(3+1) = -16 \times (x^4)/4 = -4x^4$
Integrate $-21x^2$ : Integrate the second term $-21x^2$ . Using the power rule, we add $1$ to the exponent and divide by the new exponent. $\int -21x^2 \, dx = -21 \times \frac{x^{2+1}}{2+1} = -21 \times \frac{x^3}{3} = -7x^3$
Combine Results: Combine the results of the integrals of both terms and add the constant of integration $C$ . $\int (-16x^3 - 21x^2) \, dx = -4x^4 - 7x^3 + C$