Apply the power rule for integration, integral xⁿ dx = xⁿ⁺¹n+1 + C, and integral k dx = kx + C.

Question

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Here are the solutions to the integration problems.

1.

Apply the power rule for integration, $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$ , and $\int k \,dx = kx + C$ .
$\int (x^7 + 1) \,dx = \frac{x^{7+1}}{7+1} + x + C$
Simplify the expression.
$= \frac{x^8}{8} + x + C$
The student's answer is ✅ Correct.
$\boxed{\frac{x^8}{8} + x + C}$
2.
Rewrite the integrand using exponent rules.
$\sqrt[3]{x^2} = x^{2/3}$ , so $\frac{1}{\sqrt[3]{x^2}} = x^{-2/3}$ .
$\int \frac{dx}{\sqrt[3]{x^2}} = \int x^{-2/3} \,dx$
Apply the power rule for integration, $\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$ .
Here $n = -\frac{2}{3}$ .
$= \frac{x^{-2/3 + 1}}{-2/3 + 1} + C = \frac{x^{1/3}}{1/3} + C$
Simplify the expression.
$= 3x^{1/3} + C$
The student's answer is ✅ Correct.
$\boxed{3x^{1/3} + C}$
3.
Apply the power rule for integration to each term.
$\int (3x^3 + 2x^2 - x + 5) \,dx = 3\frac{x^{3+1}}{3+1} + 2\frac{x^{2+1}}{2+1} - \frac{x^{1+1}}{1+1} + 5x + C$
Simplify each term.
$= \frac{3x^4}{4} + \frac{2x^3}{3} - \frac{x^2}{2} + 5x + C$
The student's answer is ✅ Correct.
$\boxed{\frac{3x^4}{4} + \frac{2x^3}{3} - \frac{x^2}{2} + 5x + C}$
4.
Simplify the integrand by dividing each term in the numerator by $x^2$ .
$\int \frac{x^2 + 3x^3 - 4x}{x^2} \,dx = \int \left(\frac{x^2}{x^2} + \frac{3x^3}{x^2} - \frac{4x}{x^2}\right) \,dx$ $= \int (1 + 3x - 4x^{-1}) \,dx$
Apply the power rule for integration and the rule for $\int \frac{1}{x} \,dx = \ln|x| + C$ .
$= x + 3\frac{x^{1+1}}{1+1} - 4\ln|x| + C$
Simplify the expression.
$= x + \frac{3x^2}{2} - 4\ln|x| + C$
The student's answer is ❌ Incorrect. The first term of the integrand was simplified incorrectly.
$\boxed{x + \frac{3x^2}{2} - 4\ln|x| + C}$

Apply the power rule for integration, integral xⁿ \,dx = xⁿ⁺¹n+1 + C, and integral k \,dx = kx + C.

Step-by-step solution

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

Simplify the expression.

Rewrite the integrand using exponent rules.

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

Simplify the expression.

Apply the power rule for integration to each term.

Simplify each term.

Simplify the integrand by dividing each term in the numerator by $x^2$ .

Apply the power rule for integration and the rule for $\int \frac{1}{x} \,dx = \ln|x| + C$ .

Simplify the expression.

Still stuck on this one?

More Mathematics Questions

Apply the power rule for integration, integral xⁿ \,dx = xⁿ⁺¹n+1 + C, and integral k \,dx = kx + C.

Step-by-step solution

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 ∫k dx=kx+C\int k \,dx = kx + C∫kdx=kx+C.

Simplify the expression.

Rewrite the integrand using exponent rules.

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.

Simplify the expression.

Apply the power rule for integration to each term.

Simplify each term.

Simplify the integrand by dividing each term in the numerator by x2x^2x2.

Apply the power rule for integration and the rule for ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \,dx = \ln|x| + C∫x1​dx=ln∣x∣+C.

Simplify the expression.

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 $\int k \,dx = kx + C$ .

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

Simplify the integrand by dividing each term in the numerator by $x^2$ .

Apply the power rule for integration and the rule for $\int \frac{1}{x} \,dx = \ln|x| + C$ .