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Math  /  Calculus

QuestionFind ∫(−2x4+2x−6x5+5x)dx\int\left(-2 x^{4}+\frac{2}{x}-\frac{6}{x^{5}}+5 \sqrt{x}\right) d x +C+C

Studdy Solution

STEP 1

1. We are dealing with an indefinite integral, which means we will find the antiderivative of the function.
2. The integral involves polynomial and power functions, which can be integrated term by term.
3. The constant of integration, denoted as C C , will be added at the end.

STEP 2

1. Integrate each term separately.
2. Apply the power rule for integration.
3. Simplify the expression and add the constant of integration.

STEP 3

Integrate each term separately. The given integral is:
∫(−2x4+2x−6x5+5x) dx\int \left(-2x^4 + \frac{2}{x} - \frac{6}{x^5} + 5\sqrt{x}\right) \, dx
We will integrate each term individually.

STEP 4

Apply the power rule for integration to each term. The power rule states that ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, where n≠−1n \neq -1.
1. For −2x4-2x^4: $ \int -2x^4 \, dx = -2 \cdot \frac{x^{4+1}}{4+1} = -\frac{2}{5}x^5 \]
2. For 2x\frac{2}{x}, rewrite as 2x−12x^{-1}: $ \int 2x^{-1} \, dx = 2 \ln |x| \]
3. For −6x5-\frac{6}{x^5}, rewrite as −6x−5-6x^{-5}: $ \int -6x^{-5} \, dx = -6 \cdot \frac{x^{-5+1}}{-5+1} = \frac{6}{4}x^{-4} = \frac{3}{2}x^{-4} \]
4. For 5x5\sqrt{x}, rewrite as 5x1/25x^{1/2}: $ \int 5x^{1/2} \, dx = 5 \cdot \frac{x^{1/2+1}}{1/2+1} = \frac{10}{3}x^{3/2} \]

STEP 5

Combine the results of the integrations and add the constant of integration C C :
−25x5+2ln⁡∣x∣+32x−4+103x3/2+C-\frac{2}{5}x^5 + 2 \ln |x| + \frac{3}{2}x^{-4} + \frac{10}{3}x^{3/2} + C
The integral of the function is:
−25x5+2ln⁡∣x∣+32x−4+103x3/2+C\boxed{-\frac{2}{5}x^5 + 2 \ln |x| + \frac{3}{2}x^{-4} + \frac{10}{3}x^{3/2} + C}

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