Math  /  Calculus

QuestionFind (2x21x+5x55x)dx\int\left(2 x^{2}-\frac{1}{x}+\frac{5}{x^{5}}-5 \sqrt{x}\right) d x +C+C

Studdy Solution

STEP 1

1. The integral is indefinite, which means we will find the antiderivative and add a constant of integration C C .
2. Each term in the integrand can be integrated separately using basic integration rules.

STEP 2

1. Break down the integral into separate integrals for each term.
2. Integrate each term individually.
3. Combine the results and add the constant of integration C C .

STEP 3

Break down the integral into separate integrals for each term:
(2x21x+5x55x)dx=2x2dx1xdx+5x5dx5xdx\int \left(2x^2 - \frac{1}{x} + \frac{5}{x^5} - 5\sqrt{x}\right) \, dx = \int 2x^2 \, dx - \int \frac{1}{x} \, dx + \int \frac{5}{x^5} \, dx - \int 5\sqrt{x} \, dx

STEP 4

Integrate each term individually.
For the first term, 2x2dx \int 2x^2 \, dx :
2x2dx=2x2+12+1=23x3\int 2x^2 \, dx = 2 \cdot \frac{x^{2+1}}{2+1} = \frac{2}{3}x^3

STEP 5

For the second term, 1xdx \int \frac{1}{x} \, dx :
1xdx=lnx\int \frac{1}{x} \, dx = \ln |x|

STEP 6

For the third term, 5x5dx \int \frac{5}{x^5} \, dx :
Rewrite 5x5 \frac{5}{x^5} as 5x5 5x^{-5} :
5x5dx=5x5+15+1=5x44=54x4\int 5x^{-5} \, dx = 5 \cdot \frac{x^{-5+1}}{-5+1} = 5 \cdot \frac{x^{-4}}{-4} = -\frac{5}{4}x^{-4}

STEP 7

For the fourth term, 5xdx \int 5\sqrt{x} \, dx :
Rewrite x \sqrt{x} as x1/2 x^{1/2} :
5x1/2dx=5x1/2+11/2+1=5x3/23/2=103x3/2\int 5x^{1/2} \, dx = 5 \cdot \frac{x^{1/2+1}}{1/2+1} = 5 \cdot \frac{x^{3/2}}{3/2} = \frac{10}{3}x^{3/2}

STEP 8

Combine the results and add the constant of integration C C :
(2x21x+5x55x)dx=23x3lnx54x4103x3/2+C\int \left(2x^2 - \frac{1}{x} + \frac{5}{x^5} - 5\sqrt{x}\right) \, dx = \frac{2}{3}x^3 - \ln |x| - \frac{5}{4}x^{-4} - \frac{10}{3}x^{3/2} + C
The antiderivative is:
23x3lnx54x4103x3/2+C\boxed{\frac{2}{3}x^3 - \ln |x| - \frac{5}{4}x^{-4} - \frac{10}{3}x^{3/2} + C}

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