Find the Antiderivative of f(x)= 3x^4 - square root of 8 x^8 + 5/9x^5

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We need the anti derivative of f(x) = 3x^4 - sqrt (8x^8) + (5/9)*x^5.

Int [ 3x^4 - sqrt (8x^8) + (5/9)*x^5  dx]

=> Int [ 3x^4 dx] - Int [sqrt (8x^8) dx] + Int[ (5/9)*x^5  dx]

=> (3/5)*x^5 - (sqrt 8/5)*(x^5) + (5/9*6)*x^6 + C

=> (3/5)*x^5 - (sqrt 8/5)*(x^5) + (5/54)*x^6 + C

The anti derivative is (3/5)*x^5 - (sqrt 8/5)*(x^5) + (5/54)*x^6 + C

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The antiderivative of the given function is the indefinite integral of f(x).

We'll calculate the antiderivative of the function you've provided:

f(x) = 3x^4dx - sqrt8*x^8 + 5/9x^5

We'll use the property of integral to be additive:

Int f(x)dx = Int 3x^4dx - Int sqrt8*x^8 + (5/9)*Int x^-5 dx

Int f(x)dx = 3x^5/5 - 2sqrt2*x^9/9 + (5/9)*x^-4/-4 + C

The anitderivative is  3x^5/5 - 2sqrt2*x^9/9 + (5/9)*x^-4/-4 + C.

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