Evaluate − 3 x 3 </mrow> + 2

zuzogiecwu

zuzogiecwu

Answered question

2022-05-13

Evaluate ( 3 x 3 + 2 x + cos ( x ) + cos ( 3 x ) )  d x.

Answer & Explanation

charringpq49u

charringpq49u

Beginner2022-05-14Added 23 answers

Remove parentheses.

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

Split the single integral into multiple integrals.

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

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

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

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

-(31x3dx)+2xdx+cos(x)dx+cos(3x)dx

Simplify the expression.

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

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

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

Simplify.

-3(-12x2+C)+2xdx+cos(x)dx+cos(3x)dx

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

-3(-12x2+C)+21xdx+cos(x)dx+cos(3x)dx

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

-3(-12x2+C)+2(ln(|x|)+C)+cos(x)dx+cos(3x)dx

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

-3(-12x2+C)+2(ln(|x|)+C)+sin(x)+C+cos(3x)dx

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

-3(-12x2+C)+2(ln(|x|)+C)+sin(x)+C+cos(u)13du

Combine cos(u) and 13.

-3(-12x2+C)+2(ln(|x|)+C)+sin(x)+C+cos(u)3du

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

-3(-12x2+C)+2(ln(|x|)+C)+sin(x)+C+13cos(u)du

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

-3(-12x2+C)+2(ln(|x|)+C)+sin(x)+C+13(sin(u)+C)

Simplify.

32x2+2ln(|x|)+sin(x)+13sin(u)+C

Replace all occurrences of uu with 3x3x.

32x2+2ln(|x|)+sin(x)+13sin(3x)+C

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