6.7.1 $\int\sqrt{x^2+a^2}\thinspace dx$ 🔧
$$\int\sqrt{x^2+a^2}\thinspace dx=\frac{x\cdot\sqrt{x^2+a^2}+a^2\cdot\ln\big(x+\sqrt{x^2+a^2}\big)}{2}+C$$
Proof
6.7.2 $\int \sqrt{x^2-a^2}\thinspace dx$ 🔧
$$\int\sqrt{x^2-a^2}\thinspace dx=\frac{x\cdot\sqrt{x^2-a^2}-a^2\cdot\ln\big|x+\sqrt{x^2-a^2}\big|}{2}+C$$
Proof
6.7.3 $\int\sqrt{a^2-x^2}\thinspace dx$ 🔧
$$\int\sqrt{a^2-x^2}\thinspace dx=\frac{x\cdot\sqrt{a^2-x^2}+a^2\cdot\sin^{-1}(x/a)}{2}+C$$
Proof
6.7.4 $\int x\cdot\sqrt{x^2+a^2}\thinspace dx$ 🔧
$$\int x\cdot\sqrt{x^2+a^2}\thinspace dx=$$
Proof
6.7.5 $\int x\cdot\sqrt{x^2-a^2}\thinspace dx$ 🔧
$$\int x\cdot\sqrt{x^2-a^2}\thinspace dx=$$
Proof
6.7.6 $\int x\cdot\sqrt{a^2-x^2}\thinspace dx$ 🔧
$$\int x\cdot\sqrt{a^2-x^2}\thinspace dx=$$
Proof
6.7.7 $\int x^2\cdot\sqrt{x^2+a^2}\thinspace dx$ 🔧
$$\int x^2\cdot\sqrt{x^2+a^2}\thinspace dx=\frac{x\cdot(2\cdot x^2+a^2)\cdot\sqrt{x^2+a^2}-a^4\cdot\ln\big(x+\sqrt{x^2+a^2}\big)}{8}+C$$
Proof
6.7.8 $\int x^2\cdot\sqrt{x^2-a^2}\thinspace dx$ 🔧
$$\int x^2\cdot\sqrt{x^2-a^2}\thinspace dx=\frac{x\cdot(2\cdot x^2-a^2)\cdot\sqrt{x^2-a^2}-a^4\cdot\ln\big|x+\sqrt{x^2-a^2}\big|}{8}+C$$
Proof
6.7.9 $\int x^2\cdot\sqrt{a^2-x^2}\thinspace dx$ 🔧
$$\int x^2\cdot\sqrt{a^2-x^2}\thinspace dx=\frac{x\cdot(2\cdot x^2-a^2)\cdot\sqrt{a^2-x^2}+a^4\cdot\sin^{-1}(x/a)}{8}+C$$
Proof
6.7.A $\int \sqrt{x^2+a^2}/x\thinspace dx$ 🔧
$$\int \frac{\sqrt{x^2+a^2}}{x}\thinspace dx=$$
Proof
6.7.B $\int \sqrt{x^2-a^2}/x\thinspace dx$ 🔧
$$\int \frac{\sqrt{x^2-a^2}}{x}\thinspace dx=$$
Proof
6.7.C $\int \sqrt{a^2-x^2}/x\thinspace dx$ 🔧
$$\int \frac{\sqrt{a^2-x^2}}{x}\thinspace dx=$$
Proof
6.7.D $\int \sqrt{x^2+a^2}/x^2\thinspace dx$ 🔧
$$\int \frac{\sqrt{x^2+a^2}}{x^2}\thinspace dx=\ln\big|x+\sqrt{x^2+a^2}\big|-\frac{\sqrt{x^2+a^2}}{x}+C$$
Proof
6.7.E $\int \sqrt{x^2-a^2}/x^2\thinspace dx$ 🔧
$$\int \frac{\sqrt{x^2-a^2}}{x^2}\thinspace dx=$$
Proof
6.7.F $\int \sqrt{a^2-x^2}/x^2\thinspace dx$ 🔧
$$\int \frac{\sqrt{a^2-x^2}}{x^2}\thinspace dx=-\frac{\sqrt{a^2-x^2}}{x}-\sin^{-1}\bigg(\frac{x}{a}\bigg)+C$$
Proof
6.7.G $\int 1/\sqrt{x^2+a^2}\thinspace dx$ 🔧
$$\int \frac{1}{\sqrt{x^2+a^2}}\thinspace dx=\ln\big(x+\sqrt{x^2+a^2}\big)+C$$
Proof
6.7.H $\int 1/\sqrt{x^2-a^2}\thinspace dx$ 🔧
$$\int \frac{1}{\sqrt{x^2-a^2}}\thinspace dx=$$
Proof
6.7.I $\int 1/\sqrt{a^2-x^2}\thinspace dx$ 🔧
$$\int \frac{1}{\sqrt{a^2-x^2}}\thinspace dx=\sin^{-1}\bigg(\frac{x}{a}\bigg)+C$$
Proof
6.7.J $\int x/\sqrt{x^2+a^2}\thinspace dx$ 🔧
$$\int \frac{x}{\sqrt{x^2+a^2}}\thinspace dx=$$
Proof
6.7.K $\int x/\sqrt{x^2-a^2}\thinspace dx$ 🔧
$$\int \frac{x}{\sqrt{x^2-a^2}}\thinspace dx=$$
Proof
6.7.L $\int x/\sqrt{a^2-x^2}\thinspace dx$ 🔧
$$\int \frac{x}{\sqrt{a^2-x^2}}\thinspace dx=$$
Proof
6.7.M $\int x^2/\sqrt{x^2+a^2}\thinspace dx$ 🔧
$$\int\frac{x^2}{\sqrt{x^2+a^2}}\thinspace dx=\frac{-a^2\cdot\ln\big(x+\sqrt{x^2+a^2}\big)+x\cdot\sqrt{x^2+a^2}}{2}+C$$
Proof
6.7.N $\int x^2/\sqrt{x^2-a^2}\thinspace dx$ 🔧
$$\int \frac{x^2}{\sqrt{x^2-a^2}}\thinspace dx=$$
Proof
6.7.O $\int x^2/\sqrt{a^2-x^2}\thinspace dx$ 🔧
$$\int \frac{x^2}{\sqrt{a^2-x^2}}\thinspace dx=-\frac{x\cdot\sqrt{a^2-x^2}-a^2\cdot\sin^{-1}(x/a)}{2}+C$$
Proof
6.7.P $\int 1/(x\cdot\sqrt{x^2+a^2})\thinspace dx$ 🔧
$$\int \frac{1}{x\cdot\sqrt{x^2+a^2}}\thinspace dx=$$
Proof
6.7.Q $\int 1/(x\cdot\sqrt{x^2-a^2})\thinspace dx$ 🔧
$$\int \frac{1}{x\cdot\sqrt{x^2-a^2}}\thinspace dx=\frac{1}{a}\cdot\sec^{-1}\bigg|\frac{x}{a}\bigg|+C$$
Proof
6.7.R $\int 1/(x\cdot\sqrt{a^2-x^2})\thinspace dx$ 🔧
$$\int \frac{1}{x\cdot\sqrt{a^2-x^2}}\thinspace dx=-\frac{1}{a}\cdot\ln\bigg|\frac{a+\sqrt{a^2-x^2}}{x}\bigg|+C$$
Proof
6.7.S $\int 1/(x^2\cdot\sqrt{x^2+a^2})\thinspace dx$ 🔧
$$\int \frac{1}{x^2\cdot\sqrt{x^2+a^2}}\thinspace dx=$$
Proof
6.7.T $\int 1/(x^2\cdot\sqrt{x^2-a^2})\thinspace dx$ 🔧
$$\int \frac{1}{x^2\cdot\sqrt{x^2-a^2}}\thinspace dx=$$
Proof
6.7.U $\int 1/(x^2\cdot\sqrt{a^2-x^2})\thinspace dx$ 🔧
$$\int \frac{1}{x^2\cdot\sqrt{a^2-x^2}}\thinspace dx=-\frac{\sqrt{a^2-x^2}}{a^2\cdot x}+C$$
Proof
