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上\%4:15 5月30日国 .

45 \% \pi

SampleFinalExam2_COMP 在Chrome 中打开完成 1. Find the m...

May 29, 2024

Solution by Steps

step 1

We need to find the most general antiderivative of the function

\pi x - \sin(3x) + \frac{1}{\cos^2(3x)} + (1-2x)^4

step 2

The integral of

\pi x

\frac{\pi x^2}{2}

step 3

The integral of

-\sin(3x)

\frac{1}{3} \cos(3x)

step 4

The integral of

\frac{1}{\cos^2(3x)}

\frac{1}{3} \tan(3x)

step 5

The integral of

(1-2x)^4

\frac{(1-2x)^5}{-10}

step 6

Combining all these results, we get the most general antiderivative:

step 7

\frac{\pi x^2}{2} + \frac{1}{3} \cos(3x) + \frac{1}{3} \tan(3x) - \frac{(1-2x)^5}{10} + C

, where

C

is the constant of integration

Answer

\frac{\pi x^2}{2} + \frac{1}{3} \cos(3x) + \frac{1}{3} \tan(3x) - \frac{(1-2x)^5}{10} + C

Key Concept

Antiderivative Calculation

Explanation

The antiderivative of a function is found by integrating each term separately and then combining the results, including the constant of integration.