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Step-by-step Solution

Integrate x(sin(x)-cos(x)) from 0 to pi/4

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Answer

$\frac{\left(\frac{1}{73}-\int_{0}^{\frac{2}{\sqrt{2}}}_{0}^{\frac{2}{\sqrt{2}}} x\cos\left(x\right)dx-x\cos\left(x\right)\right)}{-\cos\left(x\right)-\sin\left(x\right)}$

Step-by-step explanation

Problem to solve:

$\frac{\int_0^{\frac{\pi\:}{4}}x\left(\sin\:\left(x\right)-\cos\:\left(x\right)\right)dx}{\int_0^{\frac{\pi\:}{4}}\sin\:\left(x\right)-\cos\:\left(x\right)dx}$
1

The integral of a sum of two or more functions is equal to the sum of their integrals

$\frac{\int_{0}^{\frac{2}{\sqrt{2}}}_{\frac{2}{\sqrt{2}}}^{0}_{0}^{\frac{2}{\sqrt{2}}}\left(x\sin\left(x\right)-x\cos\left(x\right)\right)dx}{\int\sin\left(x\right)dx+\int-\cos\left(x\right)dx}$
2

The integral of a sum of two or more functions is equal to the sum of their integrals

$\frac{\left(\int_{0}^{\frac{2}{\sqrt{2}}}_{0}^{\frac{2}{\sqrt{2}}} x\sin\left(x\right)dx+\int_{0}^{\frac{2}{\sqrt{2}}}_{0}^{\frac{2}{\sqrt{2}}}-x\cos\left(x\right)dx\right)}{\int\sin\left(x\right)dx+\int-\cos\left(x\right)dx}$

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Answer

$\frac{\left(\frac{1}{73}-\int_{0}^{\frac{2}{\sqrt{2}}}_{0}^{\frac{2}{\sqrt{2}}} x\cos\left(x\right)dx-x\cos\left(x\right)\right)}{-\cos\left(x\right)-\sin\left(x\right)}$

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$\frac{\int_0^{\frac{\pi\:}{4}}x\left(\sin\:\left(x\right)-\cos\:\left(x\right)\right)dx}{\int_0^{\frac{\pi\:}{4}}\sin\:\left(x\right)-\cos\:\left(x\right)dx}$

Main topic:

Integration by parts

Used formulas:

6. See formulas

Time to solve it:

~ 0.74 seconds