INTEGRALS CONTAINING "xⁿ+aⁿ" or "xⁿ-aⁿ"

1.

$$\int\displaystyle \frac{dx}{x\left(x^{\displaystyle n}+a^{\displa... ...isplaystyle \frac{x^{\displaystyle n}}{x^{\displaystyle n}+a^{\displaystyle n}}$$

2.

$$\int\displaystyle \frac{x^{\displaystyle n-1}\,dx}{x^{\displaysty... ...displaystyle \frac{1}{n}\ln\left(x^{\displaystyle n}+a^{\displaystyle n}\right)$$

3.

$$\int\displaystyle \frac{x^{\displaystyle m}\,dx}{\left(x^{\displa... ...n}\,dx}{\left(x^{\displaystyle n}+a^{\displaystyle n}\right)^{\displaystyle r}}$$

4.

$$\int\displaystyle \frac{dx}{x^{\displaystyle m}\left(x^{\displays... ...yle m-n}\left(x^{\displaystyle n}+a^{\displaystyle n}\right)^{\displaystyle r}}$$

5.

$$\int\displaystyle \frac{dx}{x\displaystyle \sqrt{x^{\displaystyle... ...ystyle n}+a^{\displaystyle n}}+\displaystyle \sqrt{a^{\displaystyle n}}}\right)$$

6.

$$\int\displaystyle \frac{dx}{x\left(x^{\displaystyle n}-a^{\displa... ...tyle \frac{x^{\displaystyle n}-a^{\displaystyle n}}{x^{\displaystyle n}}\right)$$

7.

$$\int\displaystyle \frac{x^{\displaystyle n-1}\,dx}{x^{\displaysty... ...displaystyle \frac{1}{n}\ln\left(x^{\displaystyle n}-a^{\displaystyle n}\right)$$

8.

$$\int\displaystyle \frac{x^{\displaystyle m}\,dx}{\left(x^{\displa... ...\,dx}{\left(x^{\displaystyle n}-a^{\displaystyle n}\right)^{\displaystyle r-1}}$$

9.

$$\int\displaystyle \frac{dx}{x^{\displaystyle m}\left(x^{\displays... ...yle m}\left(x^{\displaystyle n}-a^{\displaystyle n}\right)^{\displaystyle r-1}}$$

10.

$$\int\displaystyle \frac{dx}{x\displaystyle \sqrt{x^{\displaystyle... ...splaystyle \sqrt{\displaystyle \frac{a^{\displaystyle n}}{x^{\displaystyle n}}}$$

11.

$$\begin{array}{lcl} \displaystyle \int\displaystyle \frac{x^{\disp... ...\cos\displaystyle \frac{(2k-1)\pi}{2m}+a^{\displaystyle2}\right)\\ \end{array}$$

where $0<p\leq 2m$.

12.

$$\begin{array}{lcl} \displaystyle \int\displaystyle \frac{x^{\disp... ...^{\displaystyle2m-p}}\{\ln(x-a)+(-1)^{\displaystyle p}\ln(x+a)\}\\ \end{array}$$

where $0<p\leq 2m$.

13.

$$\begin{array}{ll} \displaystyle \int\displaystyle \frac{x^{\displ... ...(-1)^{\displaystyle p-1}\ln(x+a)}{(2m+1)a^{\displaystyle2m-p+1}}\\ \end{array}$$

where $0<p\leq2m+1$.

14.

$$\begin{array}{ll} \displaystyle \int\displaystyle \frac{x^{\displa... ...} }+\displaystyle \frac{\ln(x-a)}{(2m+1)a^{\displaystyle2m-p+1}}\\ \end{array}$$

where $0<p\leq2m+1$.

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