What is the integral from x to 4-x of (x^2y+1/y) ?

The integral from x to 4-x of (x^2y+1/y) is ln|4-x|-ln|x|+4x^2(-x+2)

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integral from x to 4-x of (x^2y+1/y)

\int_{x}^{4 - x} (x^{2} y + \frac{1}{y}) d y

ln ∣ 4 - x ∣ - ln ∣ x ∣ + 4 x^{2} (- x + 2)

Solution steps

\int_{x}^{4 - x} x^{2} y + \frac{1}{y} d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{x}^{4 - x} \frac{1}{y} d y + \int_{x}^{4 - x} x^{2} y d y

\int_{x}^{4 - x} \frac{1}{y} d y = ln ∣ 4 - x ∣ - ln ∣ x ∣

\int_{x}^{4 - x} x^{2} y d y = 4 x^{2} (- x + 2)

= ln ∣ 4 - x ∣ - ln ∣ x ∣ + 4 x^{2} (- x + 2)

What is the integral from x to 4-x of (x^2y+1/y) ?
The integral from x to 4-x of (x^2y+1/y) is ln|4-x|-ln|x|+4x^2(-x+2)