Find the given integral.∫1x In x3dx

An Smith

An Smith

Answered question

2022-05-16

Find the given integral.

1x In x3dx

Answer & Explanation

alenahelenash

alenahelenash

Expert2022-06-05Added 556 answers

1xln(x3)dx

Rewrite as 1x(3ln(x))dx.

1x(3ln(x))dx

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

131x(ln(x))dx

Let u=ln(x). Then du=1xdx, so xdu=dx. Rewrite using uu and dduu.

Let u=ln(x). Find dudx.

Differentiate ln(x).

ddx[ln(x)]

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

1x

Rewrite the problem using u and du.

131udu

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

13(ln(|u|)+C)

Simplify.

13ln(|u|)+C

Replace all occurrences of u with ln(x).

13ln(ln|(x)|)+C

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