What is the integral from 0 to 1 of (2x+3)/((x+1)^2) ?

The integral from 0 to 1 of (2x+3)/((x+1)^2) is 2ln(2)+1/2

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integral from 0 to 1 of (2x+3)/((x+1)^2)

\int_{0}^{1} \frac{2 x + 3}{( x + 1 ) ^{2}} d x

2 ln (2) + \frac{1}{2}

Decimal Notation

1.88629 \dots

Solution steps

\int_{0}^{1} \frac{2 x + 3}{( x + 1 ) ^{2}} d x

Expand

\frac{2 x + 3}{( x + 1 ) ^{2}} : \frac{2 x}{( x + 1 ) ^{2}} + \frac{3}{( x + 1 ) ^{2}}

Apply the Sum Rule:

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

= \int_{0}^{1} \frac{2 x}{( x + 1 ) ^{2}} d x + \int_{0}^{1} \frac{3}{( x + 1 ) ^{2}} d x

\int_{0}^{1} \frac{2 x}{( x + 1 ) ^{2}} d x = 2 ln (2) - 1

\int_{0}^{1} \frac{3}{( x + 1 ) ^{2}} d x = \frac{3}{2}

= 2 ln (2) - 1 + \frac{3}{2}

Combine the fractions

- 1 + \frac{3}{2} : \frac{1}{2}

= 2 ln (2) + \frac{1}{2}

\int (\frac{e ^{- \frac{2 y}{x}} y ^{3}}{x ^{4}}) d x

\frac{3}{5} cos (2 x - \frac{π}{4}) + π

y = 4 π r^{2}

\frac{d}{d x} (x^{7} ln (x))

x^{2} e^{- x}, x e^{- x}

What is the integral from 0 to 1 of (2x+3)/((x+1)^2) ?
The integral from 0 to 1 of (2x+3)/((x+1)^2) is 2ln(2)+1/2