Compute 3 <mrow> 2 x 3 </mrow>

hard12bb30crg

hard12bb30crg

Answered question

2022-05-13

Compute ( 3 2 x 3 2 x + sin ( x ) + cos ( 3 x ) ) d x.

Answer & Explanation

Frida Wilkinson

Frida Wilkinson

Beginner2022-05-14Added 13 answers

Remove parentheses.

32x3-2x+sin(x)+cos(3x)dx

Split the single integral into multiple integrals.

32x3dx+-2xdx+sin(x)dx+cos(3x)dx

Since 32 is constant with respect to x, move 32 out of the integral.

321x3dx+-2xdx+sin(x)dx+cos(3x)dx

Apply basic rules of exponents.

32x-3dx+-2xdx+sin(x)dx+cos(3x)dx

By the Power Rule, the integral of x-3 with respect to x is -12x-2.

32(-12x-2+C)+-2xdx+sin(x)dx+cos(3x)dx

Since -1 is constant with respect to x, move -1 out of the integral.32(-12x-2+C)-2xdx+sin(x)dx+cos(3x)dx

Since 2 is constant with respect to x, move 2 out of the integral.

32(-12x-2+C)-(21xdx)+sin(x)dx+cos(3x)dx

Multiply 2 by -1.

32(-12x-2+C)-21xdx+sin(x)dx+cos(3x)dx

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

32(-12x-2+C)-2(ln(|x|)+C)+sin(x)dx+cos(3x)dx

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

32(-12x-2+C)-2(ln(|x|)+C)-cos(x)+C+cos(3x)dx

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

32(-12x-2+C)-2(ln(|x|)+C)-cos(x)+C+cos(u)13du

Combine cos(u) and 13.

32(-12x-2+C)-2(ln(|x|)+C)-cos(x)+C+cos(u)3du

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

32(-12x-2+C)-2(ln(|x|)+C)-cos(x)+C+13cos(u)du

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

32(-12x-2+C)-2(ln(|x|)+C)-cos(x)+C+13(sin(u)+C)

Simplify.

-34x2-2ln(|x|)-cos(x)+13sin(u)+C

Replace all occurrences of u with 3x.

-34x2-2ln(|x|)-cos(x)+13sin(3x)+C

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