# How do you find the definite integral of #(3x^3 +7) dx # from #[1, 5]#?

496

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To find the definite integral of (3x^3 + 7) with respect to (x) from (1) to (5), you first need to find the antiderivative of the function. The antiderivative of (3x^3 + 7) is (\frac{3}{4}x^4 + 7x). Then, you evaluate this antiderivative at the upper limit (5) and subtract the result when evaluated at the lower limit (1).

( \int_{1}^{5} (3x^3 + 7) , dx = \left[ \frac{3}{4}x^4 + 7x \right]_{1}^{5} = \left( \frac{3}{4}(5)^4 + 7(5) \right) - \left( \frac{3}{4}(1)^4 + 7(1) \right) = \left( \frac{3}{4}(625) + 35 \right) - \left( \frac{3}{4}(1) + 7 \right) = \left( \frac{1875}{4} + 35 \right) - \left( \frac{3}{4} + 7 \right) = \left( \frac{1875}{4} + 35 \right) - \left( \frac{3}{4} + 7 \right) = \frac{1875}{4} + 35 - \frac{3}{4} - 7 = \frac{1875}{4} - \frac{3}{4} + 35 - 7 = \frac{1872}{4} + 28 = 468 + 28 = 496. )

So, the definite integral of (3x^3 + 7) from (1) to (5) is (496).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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