Evaluate the integral: integral 7x . ln (x) dx.

Question:

Evaluate the integral:

{eq}\displaystyle \int 7x \cdot \ln (x)\ dx {/eq}.

Integral:

Let us say we have the function

{eq}\int uf\\ =u\int fdx-\int \frac{\mathrm{d} u}{\mathrm{d} x}\int fdx {/eq}. We will take the first function which is easy to differentiate

Answer and Explanation:

To find the derivative we will use the integration by parts

Let us say we have the function

{eq}\displaystyle\int uf\\ =\displaystyle u\int fdx-\int \frac{\mathrm{d} u}{\mathrm{d} x}\int fdx {/eq}

Now integrating we get

{eq}=\displaystyle\frac{\ln x7x^{2}}{2}-\int \frac{1}{x}\frac{7x^{2}}{2}dx\\ =\displaystyle\frac{7x^{2}\ln x}{2}-\frac{7x^{2}}{4}+c {/eq}


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