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How do you find the antiderivative of #f(x)=1/5x^4-3/4x-1#?

Answer 1

Isabella Ackerman

#1/25x^(5)-3/8x^(2)-x+C#

Note #(a*1/(n+1)*x^(n+1))'=a*x^n#

#int1/5x^4-3/4x-1dx#

#1/5*1/(4+1)x^(4+1)-3/4*1/(1+1)x^(1+1)-x#

#1/5*1/(5)x^(5)-3/4*1/(2)x^(2)-x+C#

#1/25x^(5)-3/8x^(2)-x+C#

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Answer 2

Isaiah Allen

To find the antiderivative of ( f(x) = \frac{1}{5}x^4 - \frac{3}{4}x - 1 ), you integrate each term separately. The antiderivative of ( x^n ) is ( \frac{x^{n+1}}{n+1} ) for all real numbers except ( n = -1 ). Using this formula:

[ \int \frac{1}{5}x^4 , dx = \frac{1}{5} \cdot \frac{x^5}{5} + C ] [ \int -\frac{3}{4}x , dx = -\frac{3}{4} \cdot \frac{x^2}{2} + C ] [ \int -1 , dx = -x + C ]

Adding these antiderivatives together:

[ \frac{1}{5} \cdot \frac{x^5}{5} - \frac{3}{4} \cdot \frac{x^2}{2} - x + C ]

Where ( C ) is the constant of integration.

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Answer from HIX Tutor

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.