7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

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Multiple Choice

Find $h\left(x\right)$ by evaluating the following indefinite integral.
$h (x) = 25 x^{4} d x$

$h (x) = 25 x^{5}$

$h (x) = 5 x^{5}$

$h (x) = 5 x^{5} + C$

$h (x) = 25 x^{5} + C$

Verified step by step guidance

Identify the integral to be evaluated: $ \int 25x^4 \, dx $. This is an indefinite integral, meaning we are looking for a function whose derivative is $ 25x^4 $.

Apply the power rule for integration, which states that $ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $, where $ C $ is the constant of integration.

In this case, $ n = 4 $, so we need to integrate $ 25x^4 $. First, factor out the constant 25 from the integral: $ 25 \int x^4 \, dx $.

Use the power rule: $ \int x^4 \, dx = \frac{x^{4+1}}{4+1} = \frac{x^5}{5} $.

Multiply the result by the constant 25: $ 25 \times \frac{x^5}{5} = 5x^5 $. Don't forget to add the constant of integration $ C $, giving the final expression: $ h(x) = 5x^5 + C $.