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Finding antiderivatives by undoing the chain rule: For each function f that follows, find a function F with the property that Fx=fx. You may have to guess and check to find such a function.

role="math" localid="1654760569311" fx=125x3

Short Answer

Expert verified

The final integrate function is151t3dt=11025x2+C

Step by step solution

01

Step 1. Given Information.

The given function is fx=125x3.

02

Step 2. Find the integration of the given function.

The given function is fx=125x3....i

F'x=fx

Adding up the numbers (1).

localid="1654761391962" 1(25x)3dx

Multiplying and dividing by 5.

155(25x)3dx...ii

Assume that25x=t distinguishing in terms of x.

5xdx=dt

Replace the value in equation (2).

151t3dt

Now integrate the obtained function.

localid="1654761744441" 151t3dt=15t3dt151t3dt=15[t]3+13+1+C151t3dt=15×12t-2+C151t3dt=110t2+C

C denotes the integration constant.

Adding the value of t to the equation above.

151t3dt=11025x2+C

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