6.4.1 $\int\sin^n(x)\thinspace dx$ 🔧
$$\int\sin^n(x)\thinspace dx=-\frac{1}{n}\cdot \sin^{n-1}(x)\cdot\cos(x)+\frac{n-1}{n}\cdot\int\sin^{n-2}(x)\thinspace dx$$
Proof
Example
$$\int\sin^3(x)\thinspace dx$$
6.4.2 $\int\cos^n(x)\thinspace dx$ 🔧
$$\int\cos^n(x)\thinspace dx=\frac{1}{n}\cdot \cos^{n-1}(x)\cdot\sin(x)+\frac{n-1}{n}\cdot\int\cos^{n-2}(x)\thinspace dx$$
Proof
Example
$$\int\cos^3(x)\thinspace dx$$
6.4.3 $\int\sec^n(x)\thinspace dx$ 🔧
$$\int\sec^n(x)\thinspace dx=\frac{\sec^{n-2}(x)\cdot\tan(x)}{n-1}+\frac{n-2}{n-1}\cdot\int\sec^{n-2}(x)\thinspace dx$$
Proof
Example
$$\int\sec^3(x)\thinspace dx$$
6.4.4 $\int\csc^n(x)\thinspace dx$ 🔧
$$\int\csc^n(x)\thinspace dx=-\frac{\csc^{n-2}(x)\cdot\cot(x)}{n-1}+\frac{n-2}{n-1}\cdot\int\csc^{n-2}(x)\thinspace dx$$
Proof
Example
$$\int\csc^3(x)\thinspace dx$$
6.4.5 $\int\tan^n(x)\thinspace dx$ 🔧
$$\int\tan^n(x)\thinspace dx=\frac{\tan^{n-1}(x)}{n-1}-\int\tan^{n-2}(x)\thinspace dx$$
Proof
Example
$$\int\tan^3(x)\thinspace dx$$
6.4.6 $\int\cot^n(x)\thinspace dx$ 🔧
$$\int\cot^n(x)\thinspace dx=-\frac{\cot^{n-1}(x)}{n-1}-\int\cot^{n-2}(x)\thinspace dx$$
Proof
Example
$$\int\cot^3(x)\thinspace dx$$
6.4.7 $\int\sec^n(a\cdot x)\cdot\tan(a\cdot x)\thinspace dx$ 🔧
$$\int\sec^n(a\cdot x)\cdot\tan(a\cdot x)\thinspace dx=\frac{1}{a\cdot n}\cdot\sec^n(a\cdot x)+C$$
Proof
6.4.8 $\int\csc^n(a\cdot x)\cdot\cot(a\cdot x)\thinspace dx$ 🔧
$$\int\csc^n(a\cdot x)\cdot\cot(a\cdot x)\thinspace dx=-\frac{1}{a\cdot n}\cdot\csc^n(a\cdot x)+C$$
Proof
6.4.9 $\int 1/(1\pm sin(a\cdot x))\thinspace dx$ 🔧
$$\int\frac{1}{1\pm sin(a\cdot x)}\thinspace dx=\mp\frac{1}{a}\cdot\tan\bigg(\frac{\pi}{4}\mp\frac{a\cdot x}{2}\bigg)+C$$
Proof
6.4.A $\int 1/(1\pm cos(a\cdot x))\thinspace dx$ 🔧
$$\int\frac{1}{1\pm cos(a\cdot x)}\thinspace dx=\pm\frac{1}{a}\cdot\cot\bigg(\frac{a\cdot x}{2}\bigg)+C$$
Proof
6.4.B $\int\sin(a\cdot x)\cdot\sin(b\cdot x)\thinspace dx$ 🔧
$$\int\sin(a\cdot x)\cdot\sin(b\cdot x)\thinspace dx=\frac{x\cdot\sin(a-b)}{2\cdot(a-b)}-\frac{x\cdot\sin(a+b)}{2\cdot(a+b)}+C$$
Proof
6.4.C $\int\sin(a\cdot x)\cdot\cos(b\cdot x)\thinspace dx$ 🔧
$$\int\sin(a\cdot x)\cdot\cos(b\cdot x)\thinspace dx=-\frac{x\cdot\cos(a+b)}{2\cdot(a+b)}-\frac{x\cdot\cos(a-b)}{2\cdot(a-b)}+C$$
Proof
6.4.D $\int\cos(a\cdot x)\cdot\cos(b\cdot x)\thinspace dx$ 🔧
$$\int\cos(a\cdot x)\cdot\cos(b\cdot x)\thinspace dx=\frac{x\cdot\sin(a-b)}{2\cdot(a-b)}+\frac{x\cdot\sin(a+b)}{2\cdot(a+b)}+C$$
Proof
6.4.E $\sin^m(x)\cdot\cos^n(x)$ (sine reduction) 🔧
$$\sin^m(x)\cdot\cos^n(x)=-\frac{\sin^{m-1}(x)\cdot\cos^{n+1}(x)}{m+n}+\frac{m-1}{m+n}\cdot\int\sin^{m-2}(x)\cdot\cos^n(x)\thinspace dx+C$$
Proof
6.4.F $\sin^m(x)\cdot\cos^n(x)$ (cosine reduction) 🔧
$$\sin^m(x)\cdot\cos^n(x)=\frac{\sin^{m+1}(x)\cdot\cos^{n-1}(x)}{m+n}+\frac{n-1}{m+n}\cdot\int\sin^m(x)\cdot\cos^{n-2}(x)\thinspace dx+C$$
Proof
