What is the integral of (bx^a)/(1+x^{a+1)} ?

The integral of (bx^a)/(1+x^{a+1)} is b 1/(a+1)ln|1+x^{a+1}|+C

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integral of (bx^a)/(1+x^{a+1)}

\int \frac{b x ^{a}}{1 + x ^{a + 1}} d x

b \frac{1}{a + 1} ln 1 + x^{a + 1} + C

Solution steps

\int \frac{b x ^{a}}{1 + x ^{a + 1}} d x

Take the constant out:

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

= b \cdot \int \frac{x ^{a}}{1 + x ^{a + 1}} d x

Apply u-substitution

: \int \frac{1}{u ( a + 1 )} d u

= b \cdot \int \frac{1}{u ( a + 1 )} d u

Take the constant out:

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

= b \frac{1}{a + 1} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= b \frac{1}{a + 1} ln ∣ u ∣

Substitute back

u = 1 + x^{a + 1}

= b \frac{1}{a + 1} ln 1 + x^{a + 1}

Add a constant to the solution

= b \frac{1}{a + 1} ln 1 + x^{a + 1} + C

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What is the integral of (bx^a)/(1+x^{a+1)} ?
The integral of (bx^a)/(1+x^{a+1)} is b 1/(a+1)ln|1+x^{a+1}|+C