Evaluate the indefinite Integral: Integral(4x3 - 6x + 5)dx

Question

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To evaluate the indefinite integral ∫ (4x³ − 6x + 5) dx, we will integrate each term separately using the power rule for integration, ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, and the constant rule, ∫ k dx = kx + C. Step 1: Integrate the first term, 4x³. ∫ 4x³ dx = 4 ∫ x³ dx = 4 ((x³⁺¹)/(3+1)) + C₁ = 4 ((x⁴)/(4)) + C₁ = x⁴ + C₁ Step 2: Integrate the second term, −6x. ∫ −6x dx = −6 ∫ x¹ dx = −6 ((x¹⁺¹)/(1+1)) + C₂ = −6 ((x²)/(2)) + C₂ = −3x² + C₂ Step 3: Integrate the third term, 5. ∫ 5 dx = 5x + C₃ Step 4: Combine the results from each term and add a single constant of integration, C, which represents the sum of C₁, C₂, and C₃. ∫ (4x³ − 6x + 5) dx = x⁴ − 3x² + 5x + C The indefinite integral is: x⁴ − 3x² + 5x + C

Evaluate the indefinite Integral: Integral(4x3 - 6x + 5)dx

Answer

Step-by-step solution

Integrate the first term, $4x^3$ .

Integrate the second term, $-6x$ .

Integrate the third term, 5.

Combine the results from each term and add a single constant of integration, $C$ , which represents the sum of $C_1, C_2,$ and $C_3$ .

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Evaluate the indefinite Integral: Integral(4x3 - 6x + 5)dx

Answer

Step-by-step solution

Integrate the first term, 4x34x^34x3.

Integrate the second term, −6x-6x−6x.

Integrate the third term, 5.

Combine the results from each term and add a single constant of integration, CCC, which represents the sum of C1,C2,C_1, C_2,C1​,C2​, and C3C_3C3​.

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Integrate the first term, $4x^3$ .

Integrate the second term, $-6x$ .

Combine the results from each term and add a single constant of integration, $C$ , which represents the sum of $C_1, C_2,$ and $C_3$ .