Question

Graphing calculator needed, quite simple to solve if you have one. I just do not have one. Thank you!

Answer 1

`(d)/(dx)\int x^3\ln x^2dx`

METHOD : We will use integration by part

`\begin{gathered} \int udv=uv-\int vdu \\ \end{gathered}`

I will modify the question by putting "1" before the In(x^2) function. See modified question in equation tab below

`\begin{gathered} \int1\ln(x^2)dx \\ u=\ln(x^2) \\ (du)/(dx)=(2x)/(x^2)=(2)/(x) \\ \\ dv=1 \\ v=\int1dx \\ v=x \\ \end{gathered}`
`\begin{gathered} \int1\ln(x^2)=x\ln x^2-\int x(2dx)/(x) \\ \\ =x\ln x^2-\int2dx \\ \\ =x\ln x^2-2x \end{gathered}`

`\begin{gathered} \int_2^(x^3)[x\ln x^2-2x]dx \\ \\ [x^3\ln(x^3)^2-2(x^(`3))]-[2\ln2^2-2(2)] \\ x^3\ln x^6-2x^3-2\ln4-4 \end{gathered}`