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Solving integrals/ x*ln(x^4)dx

Integral of x*ln(x^4)dx dx

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The solution

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∫01xlog⁡(x4) dx\int\limits_{0}^{1} x \log{\left(x^{4} \right)}\, dx 
Integral(x*log(x^4), (x, 0, 1))
Detail solution

  1. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=log⁡(x4)u{\left(x \right)} = \log{\left(x^{4} \right)} and let dv⁡(x)=x\operatorname{dv}{\left(x \right)} = x.

    Then du⁡(x)=4x\operatorname{du}{\left(x \right)} = \frac{4}{x}.

    To find v(x)v{\left(x \right)}:

    1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

    Now evaluate the sub-integral.

  2. The integral of a constant times a function is the constant times the integral of the function:

    ∫2x dx=2∫x dx\int 2 x\, dx = 2 \int x\, dx

    1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

    So, the result is: x2x^{2}

  3. Now simplify:

    x2(log⁡(x4)−2)2\frac{x^{2} \left(\log{\left(x^{4} \right)} - 2\right)}{2}

  4. Add the constant of integration:

    x2(log⁡(x4)−2)2+constant\frac{x^{2} \left(\log{\left(x^{4} \right)} - 2\right)}{2}+ \mathrm{constant}

The answer is:

x2(log⁡(x4)−2)2+constant\frac{x^{2} \left(\log{\left(x^{4} \right)} - 2\right)}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
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∫xlog⁡(x4) dx=C+x2log⁡(x4)2−x2\int x \log{\left(x^{4} \right)}\, dx = C + \frac{x^{2} \log{\left(x^{4} \right)}}{2} - x^{2}
Simplify
The graph
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Plot the graph f(x)
The answer [src] 
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Numerical answer [src] 
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    Use the examples entering the upper and lower limits of integration.

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