Evaluate the integral, for k,n in N: I = integral
Evaluate the integral, for $k,n\in\mathbb N$: $$ I = \int \frac{x^n}{1+x^{2k}}\,dx. $$
Explanation of Method
The method depends on how $n$ compares to $2k$. The cleanest case is $n=2k-1$, where the numerator is exactly (a multiple of) the derivative of the denominator, giving a single logarithm. Other odd $n$ reduce via $u=x^2$; even $n$ generally require full partial fractions; and $n\geq2k$ requires polynomial division first.
- Handle the special case $n=2k-1$.
- Reduce other odd $n$ via $u=x^2$.
- Divide first when $n\geq2k$.
- Verify against worked examples.
Step 1 · The clean case $n=2k-1$
Since $\dfrac{d}{dx}\left(1+x^{2k}\right)=2kx^{2k-1}$:
$$ \int\frac{x^{2k-1}}{1+x^{2k}}\,dx = \frac1{2k}\ln\left(1+x^{2k}\right) + C. $$
Step 2 · Odd $n=2m+1<2k-1$
Substituting $u=x^2$, $du=2x\,dx$:
$$ \int\frac{x^{2m+1}}{1+x^{2k}}\,dx = \frac12\int\frac{u^m}{1+u^k}\,du, $$
which reduces the exponents but generally still needs partial fractions unless $m=k-1$ (the case from Step 1 in disguise).
Step 3 · Even $n$, and $n\geq2k$
For even $n<2k$, no derivative-matching shortcut exists in general, and the integral is evaluated via the full partial fraction decomposition of $1/(1+x^{2k})$ from Problem 276, multiplied by $x^n$. For $n\geq2k$, polynomial long division first writes $x^n=(1+x^{2k})Q(x)+R(x)$ with $\deg R<2k$, reducing to $\int Q(x)\,dx$ (a polynomial) plus a proper-fraction integral handled by the same cases.
Step 4 · Worked examples (all verified by differentiation)
$$ \int\frac{x\,dx}{1+x^2}=\frac12\ln(1+x^2)+C, \qquad \int\frac{dx}{1+x^2}=\arctan x+C, $$
$$ \int\frac{x^2\,dx}{1+x^2}=x-\arctan x+C, \qquad \int\frac{x^3\,dx}{1+x^4}=\frac14\ln(1+x^4)+C, $$
$$ \int\frac{x^3\,dx}{1+x^2}=\int\left(x-\frac{x}{1+x^2}\right)dx=\frac{x^2}2-\frac12\ln(1+x^2)+C. $$
Summary Table
| Case | Method |
|---|---|
| $n=2k-1$ | Direct log via derivative match |
| $n$ odd, $n<2k-1$ | Substitute $u=x^2$, then reduce |
| $n$ even, $n<2k$ | Full partial fractions of $1/(1+x^{2k})$ |
| $n\geq2k$ | Polynomial division first |
Additional Notes
- All five worked examples were verified by differentiation: e.g. $\tfrac{d}{dx}\left[\tfrac{x^2}2-\tfrac12\ln(1+x^2)\right]=x-\tfrac{x}{1+x^2}=\tfrac{x(1+x^2)-x}{1+x^2}=\tfrac{x^3}{1+x^2}$, matching the integrand.
- Like Problems 276 and 277, this is a survey of methods rather than a single closed form — the exact antiderivative depends on the specific $n,k$ pair.
Result
The solution depends on the case: