Question #5231

f(x)=fourth root of x^3 + cubed root of x^4. Find the most general antiderivative of the function. (Check your answer by differentiation)

Expert's answer

Let f(x)=fourth root of x^3 + cubed root of x^4

= root[4]{x^3} +

root[3]{x^4} =

= x^{3/4} + x^{4/3}

Find the most general

antiderivative of the function.

(Check your answer by

differentiation)

Solution.

The antiderivative of f is

F(x) =

integral f(x) dx =

= integral ( root[4]{x^3} + root[3]{x^4} ) dx =

= integral ( x^{3/4} + x^{4/3} ) dx =

= x^{3/4+1} / (3/4 + 1)

+ x^{4/3+1} / (4/3 + 1) + C

= 4 x^{7/4} / 7 + 3 x^{7/3} / 7 +

C

Let us verify that that F'(x) = f(x).

F'(x) = (7 x^{7/4} / 4 +

7 x^{7/3} / 3 + C )' =

= 4/7 * 7/4 x^{7/4-1} + 3/7 * 7/3 x^{7/3-1}

=

= x^{3/4} + x^{4/3} =

= f(x).

= root[4]{x^3} +

root[3]{x^4} =

= x^{3/4} + x^{4/3}

Find the most general

antiderivative of the function.

(Check your answer by

differentiation)

Solution.

The antiderivative of f is

F(x) =

integral f(x) dx =

= integral ( root[4]{x^3} + root[3]{x^4} ) dx =

= integral ( x^{3/4} + x^{4/3} ) dx =

= x^{3/4+1} / (3/4 + 1)

+ x^{4/3+1} / (4/3 + 1) + C

= 4 x^{7/4} / 7 + 3 x^{7/3} / 7 +

C

Let us verify that that F'(x) = f(x).

F'(x) = (7 x^{7/4} / 4 +

7 x^{7/3} / 3 + C )' =

= 4/7 * 7/4 x^{7/4-1} + 3/7 * 7/3 x^{7/3-1}

=

= x^{3/4} + x^{4/3} =

= f(x).

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