Math  /  Calculus

QuestionThis is similar to Section 4.5 Problem 6:
Determine the indefinite integral 4+8x3xdx\int \frac{-4+8 x^{3}}{x} d x by algebraic manipulation. Assume x>0x>0 when ln(x)\ln (x) appears. Use capital C for the free constant.
Answer: \square

Studdy Solution

STEP 1

1. The integral is indefinite, meaning we are looking for a general antiderivative.
2. The expression can be simplified by dividing each term in the numerator by x x .
3. The natural logarithm function ln(x)\ln(x) may appear, and we assume x>0 x > 0 .

STEP 2

1. Simplify the integrand by dividing each term by x x .
2. Integrate each term separately.
3. Combine the results and include the constant of integration C C .

STEP 3

Simplify the integrand by dividing each term in the numerator by x x :
4+8x3xdx=(4x+8x3x)dx \int \frac{-4 + 8x^3}{x} \, dx = \int \left( \frac{-4}{x} + \frac{8x^3}{x} \right) \, dx
Simplify each term:
=(4x+8x2)dx = \int \left( -\frac{4}{x} + 8x^2 \right) \, dx

STEP 4

Integrate each term separately:
For 4x-\frac{4}{x}, the integral is:
4xdx=4lnx \int -\frac{4}{x} \, dx = -4 \ln |x|
For 8x28x^2, the integral is:
8x2dx=8x33=83x3 \int 8x^2 \, dx = 8 \cdot \frac{x^3}{3} = \frac{8}{3} x^3

STEP 5

Combine the results and include the constant of integration C C :
4+8x3xdx=4lnx+83x3+C \int \frac{-4 + 8x^3}{x} \, dx = -4 \ln |x| + \frac{8}{3} x^3 + C
The indefinite integral is:
4lnx+83x3+C \boxed{-4 \ln |x| + \frac{8}{3} x^3 + C}

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