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Math  /  Calculus

Question∫−11(5x8−9)dx\int_{-1}^{1}\left(5 x^{8}-9\right) d x ∫−11(5x8−9)dx=−1529\int_{-1}^{1}\left(5 x^{8}-9\right) d x=-\frac{152}{9} (Type an integer or a simplified fraction.)

Studdy Solution

STEP 1

1. The integral is definite, with limits from −1-1 to 11.
2. The function 5x8−95x^8 - 9 is continuous over the interval [−1,1][-1, 1].
3. We will apply the Fundamental Theorem of Calculus to evaluate the integral.

STEP 2

1. Integrate the function 5x8−95x^8 - 9.
2. Evaluate the antiderivative at the upper and lower limits.
3. Subtract the evaluated values to find the definite integral.

STEP 3

First, find the antiderivative of the function 5x8−95x^8 - 9.
The antiderivative of 5x85x^8 is:
∫5x8 dx=59x9+C\int 5x^8 \, dx = \frac{5}{9}x^9 + C
The antiderivative of −9-9 is:
∫−9 dx=−9x+C\int -9 \, dx = -9x + C
Thus, the antiderivative of the entire function is:
F(x)=59x9−9x+CF(x) = \frac{5}{9}x^9 - 9x + C

STEP 4

Evaluate the antiderivative F(x)F(x) at the upper limit x=1x = 1 and the lower limit x=−1x = -1.
At x=1x = 1:
F(1)=59(1)9−9(1)=59−9=59−819=−769F(1) = \frac{5}{9}(1)^9 - 9(1) = \frac{5}{9} - 9 = \frac{5}{9} - \frac{81}{9} = -\frac{76}{9}
At x=−1x = -1:
F(−1)=59(−1)9−9(−1)=−59+9=−59+819=769F(-1) = \frac{5}{9}(-1)^9 - 9(-1) = -\frac{5}{9} + 9 = -\frac{5}{9} + \frac{81}{9} = \frac{76}{9}

STEP 5

Subtract the value of the antiderivative at the lower limit from the value at the upper limit:
∫−11(5x8−9) dx=F(1)−F(−1)=−769−769=−1529\int_{-1}^{1} (5x^8 - 9) \, dx = F(1) - F(-1) = -\frac{76}{9} - \frac{76}{9} = -\frac{152}{9}
The value of the definite integral is:
−1529-\frac{152}{9}

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