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integral of 1/((x+1)sqrt(x^2+2x+3))

Solution

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

- \frac{1}{2 2} (ln \frac{x ^{2} + 2 x + 3}{2} + 1 - ln \frac{x ^{2} + 2 x + 3}{2} - 1) + C

Solution steps

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

Complete the square

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

= \int \frac{1}{( x + 1 ) ( x + 1 ) ^{2} + 2} d x

Apply u-substitution

: \int \frac{1}{u u ^{2} + 2} d u

= \int \frac{1}{u u ^{2} + 2} d u

Apply u-substitution

: \int \frac{1}{2 v v + 2} d v

= \int \frac{1}{2 v v + 2} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{v v + 2} d v

Apply u-substitution

: \int \frac{2}{w ^{2} - 2} d w

= \frac{1}{2} \cdot \int \frac{2}{w ^{2} - 2} d w

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot 2 \cdot \int \frac{1}{w ^{2} - 2} d w

Apply Integral Substitution

= \frac{1}{2} \cdot 2 \cdot \int \frac{1}{2 ( t ^{2} - 1 )} d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot 2 \cdot \frac{1}{2} \cdot \int \frac{1}{t ^{2} - 1} d t

Factor

t^{2} - 1 : - (- t^{2} + 1)

= \frac{1}{2} \cdot 2 \cdot \frac{1}{2} \cdot \int \frac{1}{- ( - t ^{2} + 1 )} d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot 2 \cdot \frac{1}{2} (- \int \frac{1}{- t ^{2} + 1} d t)

Use the common integral:

\int \frac{1}{- t ^{2} + 1} d t = \frac{ln ∣ t + 1 ∣}{2} - \frac{ln ∣ t - 1 ∣}{2}

= \frac{1}{2} \cdot 2 \cdot \frac{1}{2} (- (\frac{ln ∣ t + 1 ∣}{2} - \frac{ln ∣ t - 1 ∣}{2}))

Substitute back

= \frac{1}{2} \cdot 2 \cdot \frac{1}{2} - \frac{ln \frac{( x + 1 ) ^{2} + 2}{2} + 1}{2} - \frac{ln \frac{( x + 1 ) ^{2} + 2}{2} - 1}{2}

Simplify

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

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

Add a constant to the solution

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

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