Solve the following integrals: a) integral [7x3 + 2x + 1] dx b) integral [7x3 + 2x] dx c) integral [5x2 + x]…

Question

Solve the following integrals: a) integral [7x3 + 2x + 1] dx b) integral [7x3 + 2x] dx c) integral [5x2 + x] dx

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the integrals: a) For the integral ∫ [7x³ + 2x + 1] dx: Step 1: Apply the power rule for integration, ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, and the constant rule, ∫ k dx = kx + C. ∫ 7x³ dx = 7 (x³⁺¹)/(3+1) = 7 (x⁴)/(4); ∫ 2x dx = 2 (x¹⁺¹)/(1+1) = 2 (x²)/(2) = x²; ∫ 1 dx = x Step 2: Combine the results and add the constant of integration, C. ∫ [7x³ + 2x + 1] dx = 7/4x⁴ + x² + x + C The final answer is 7/4x⁴ + x² + x + C. b) For the integral ∫ [7x³ + 2x] dx: Step 1: Apply the power rule for integration to each term. ∫ 7x³ dx = 7 (x³⁺¹)/(3+1) = 7 (x⁴)/(4); ∫ 2x dx = 2 (x¹⁺¹)/(1+1) = 2 (x²)/(2) = x² Step 2: Combine the results and add the constant of integration, C. ∫ [7x³ + 2x] dx = 7/4x⁴ + x² + C The final answer is 7/4x⁴ + x² + C. c) For the integral ∫ [5x² + x] dx: Step 1: Apply the power rule for integration to each term. ∫ 5x² dx = 5 (x²⁺¹)/(2+1) = 5 (x³)/(3); ∫ x dx = (x¹⁺¹)/(1+1) = (x²)/(2) Step 2: Combine the results and add the constant of integration, C. ∫ [5x² + x] dx = 5/3x³ + 1/2x² + C The final answer is 5/3x³ + 1/2x² + C. d) For the integral ∫ [8x³ + 2] dx: Step 1: Apply the power rule and constant rule for integration to each term. ∫ 8x³ dx = 8 (x³⁺¹)/(3+1) = 8 (x⁴)/(4) = 2x⁴; ∫ 2 dx = 2x Step 2: Combine the results and add the constant of integration, C. ∫ [8x³ + 2] dx = 2x⁴ + 2x + C The final answer is 2x⁴ + 2x + C.

Solve the following integrals: a) integral [7x3 + 2x + 1] \,dx b) integral [7x3 + 2x] \,dx c) integral [5x2 + x] \,dx

Answers

Step-by-step solution

Part a· 2 steps

Apply the power rule for integration, $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$ , and the constant rule, $\int k \,dx = kx + C$ .

Combine the results and add the constant of integration, $C$ .

Part b· 2 steps
$\frac{7}{4}x^4 + x^2 + C$

Apply the power rule for integration to each term.

Combine the results and add the constant of integration, $C$ .

Part c· 2 steps
$\frac{5}{3}x^3 + \frac{1}{2}x^2 + C$

Apply the power rule for integration to each term.

Combine the results and add the constant of integration, $C$ .

Part d· 2 steps
$2x^4 + 2x + C$

Apply the power rule and constant rule for integration to each term.

Combine the results and add the constant of integration, $C$ .

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Solve the following integrals: a) integral [7x3 + 2x + 1] \,dx b) integral [7x3 + 2x] \,dx c) integral [5x2 + x] \,dx

Answers

Step-by-step solution

Part a· 2 steps

Apply the power rule for integration, ∫xn dx=xn+1n+1+C\int x^n \,dx = \frac{x^{n+1}}{n+1} + C∫xndx=n+1xn+1​+C, and the constant rule, ∫k dx=kx+C\int k \,dx = kx + C∫kdx=kx+C.

Combine the results and add the constant of integration, CCC.

Part b· 2 steps74x4+x2+C\frac{7}{4}x^4 + x^2 + C47​x4+x2+C

Apply the power rule for integration to each term.

Combine the results and add the constant of integration, CCC.

Part c· 2 steps53x3+12x2+C\frac{5}{3}x^3 + \frac{1}{2}x^2 + C35​x3+21​x2+C

Apply the power rule for integration to each term.

Combine the results and add the constant of integration, CCC.

Part d· 2 steps2x4+2x+C2x^4 + 2x + C2x4+2x+C

Apply the power rule and constant rule for integration to each term.

Combine the results and add the constant of integration, CCC.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Apply the power rule for integration, $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$ , and the constant rule, $\int k \,dx = kx + C$ .

Combine the results and add the constant of integration, $C$ .

Part b· 2 steps
$\frac{7}{4}x^4 + x^2 + C$

Combine the results and add the constant of integration, $C$ .

Part c· 2 steps
$\frac{5}{3}x^3 + \frac{1}{2}x^2 + C$

Combine the results and add the constant of integration, $C$ .

Part d· 2 steps
$2x^4 + 2x + C$

Combine the results and add the constant of integration, $C$ .