Find the integral: 3 x 3 </mrow> − x 2 −

Iyana Macdonald

Iyana Macdonald

Answered question

2022-05-11

Find the integral: ( 3 x 3 x 2 1 x + sin ( 3 x ) ) d x.

Answer & Explanation

Kyler Crawford

Kyler Crawford

Beginner2022-05-12Added 16 answers

Remove parentheses.

3x3-x2-1x+sin(3x)dx

Split the single integral into multiple integrals.

3x3dx+-x2dx+-1xdx+sin(3x)dx

Since 3 is constant with respect to x, move 3 out of the integral.

3x3dx+-x2dx+-1xdx+sin(3x)dx

By the Power Rule, the integral of x3 with respect to x is 14x4.

3(14x4+C)+-x2dx+-1xdx+sin(3x)dx

Combine 14 and x4.

3(x44+C)+-x2dx+-1xdx+sin(3x)dx

Since -1 is constant with respect to x, move -1 out of the integral.

3(x44+C)-x2dx+-1xdx+sin(3x)dx

Since 12 is constant with respect to x, move 12 out of the integral.

3(x44+C)-(12xdx)+-1xdx+sin(3x)dx

By the Power Rule, the integral of x with respect to x is 12x2.

3(x44+C)-12(12x2+C)+-1xdx+sin(3x)dx

Since -1 is constant with respect to x, move -1 out of the integral.

3(x44+C)-12(12x2+C)-1xdx+sin(3x)dx

The integral of 1x with respect to x is ln(|x|).

3(x44+C)-12(12x2+C)-(ln(|x|)+C)+sin(3x)dx

Let u=3x. Then du=3dx, so 13du=dx. Rewrite using u and du.

3(x44+C)-12(12x2+C)-(ln(|x|)+C)+sin(u)13du

Combine sin(u) and 13.

3(x44+C)-12(12x2+C)-(ln(|x|)+C)+sin(u)3du

Since 13 is constant with respect to u, move 13 out of the integral.

3(x44+C)-12(12x2+C)-(ln(|x|)+C)+13sin(u)du

The integral of sin(u) with respect to u is -cos(u).

3(x44+C)-12(12x2+C)-(ln(|x|)+C)+13(-cos(u)+C)

Simplify.

3x44-x24-ln(|x|)-cos(u)3+C

Replace all occurrences of u with 3x.

3x44-x24-ln(|x|)-cos(3x)3+C

Reorder terms.

34x4-14x2-ln(|x|)-13cos(3x)+C

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