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Math  /  Calculus

QuestionEvaluate the integral. (Remember the constant of integration.) ∫(3x5−x−7+1)dx\int\left(3 x^{5}-x^{-7}+1\right) d x

Studdy Solution

STEP 1

1. The integral is a polynomial function.
2. We will apply the power rule for integration to each term separately.
3. A constant of integration will be added at the end.

STEP 2

1. Apply the power rule to integrate each term.
2. Simplify the expression.
3. Add the constant of integration.

STEP 3

Apply the power rule for integration to each term. The power rule states that for any term axn ax^n , the integral is an+1xn+1 \frac{a}{n+1}x^{n+1} , provided n≠−1 n \neq -1 .
For the first term 3x5 3x^5 :
∫3x5 dx=35+1x5+1=36x6=12x6 \int 3x^5 \, dx = \frac{3}{5+1}x^{5+1} = \frac{3}{6}x^6 = \frac{1}{2}x^6
For the second term −x−7 -x^{-7} :
∫−x−7 dx=−1−7+1x−7+1=−1−6x−6=16x−6 \int -x^{-7} \, dx = \frac{-1}{-7+1}x^{-7+1} = \frac{-1}{-6}x^{-6} = \frac{1}{6}x^{-6}
For the third term 1 1 :
∫1 dx=x \int 1 \, dx = x

STEP 4

Combine the results of the integrals:
∫(3x5−x−7+1) dx=12x6+16x−6+x \int \left(3x^5 - x^{-7} + 1\right) \, dx = \frac{1}{2}x^6 + \frac{1}{6}x^{-6} + x

STEP 5

Add the constant of integration C C :
∫(3x5−x−7+1) dx=12x6+16x−6+x+C \int \left(3x^5 - x^{-7} + 1\right) \, dx = \frac{1}{2}x^6 + \frac{1}{6}x^{-6} + x + C
The evaluated integral is:
12x6+16x−6+x+C \boxed{\frac{1}{2}x^6 + \frac{1}{6}x^{-6} + x + C}

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