Integralen av $$$- 6 \operatorname{asin}{\left(5 x \right)}$$$

Bestäm $$$\int \left(- 6 \operatorname{asin}{\left(5 x \right)}\right)\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=-6$$$ och $$$f{\left(x \right)} = \operatorname{asin}{\left(5 x \right)}$$$:

$${\color{red}{\int{\left(- 6 \operatorname{asin}{\left(5 x \right)}\right)d x}}} = {\color{red}{\left(- 6 \int{\operatorname{asin}{\left(5 x \right)} d x}\right)}}$$

Låt $$$u=5 x$$$ vara.

Då $$$du=\left(5 x\right)^{\prime }dx = 5 dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{5}$$$.

Integralen kan omskrivas som

$$- 6 {\color{red}{\int{\operatorname{asin}{\left(5 x \right)} d x}}} = - 6 {\color{red}{\int{\frac{\operatorname{asin}{\left(u \right)}}{5} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{5}$$$ och $$$f{\left(u \right)} = \operatorname{asin}{\left(u \right)}$$$:

$$- 6 {\color{red}{\int{\frac{\operatorname{asin}{\left(u \right)}}{5} d u}}} = - 6 {\color{red}{\left(\frac{\int{\operatorname{asin}{\left(u \right)} d u}}{5}\right)}}$$

För integralen $$$\int{\operatorname{asin}{\left(u \right)} d u}$$$, använd partiell integration $$$\int \operatorname{\omega} \operatorname{dv} = \operatorname{\omega}\operatorname{v} - \int \operatorname{v} \operatorname{d\omega}$$$.

Låt $$$\operatorname{\omega}=\operatorname{asin}{\left(u \right)}$$$ och $$$\operatorname{dv}=du$$$.

Då gäller $$$\operatorname{d\omega}=\left(\operatorname{asin}{\left(u \right)}\right)^{\prime }du=\frac{du}{\sqrt{1 - u^{2}}}$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{1 d u}=u$$$ (stegen kan ses »).

Alltså,

$$- \frac{6 {\color{red}{\int{\operatorname{asin}{\left(u \right)} d u}}}}{5}=- \frac{6 {\color{red}{\left(\operatorname{asin}{\left(u \right)} \cdot u-\int{u \cdot \frac{1}{\sqrt{1 - u^{2}}} d u}\right)}}}{5}=- \frac{6 {\color{red}{\left(u \operatorname{asin}{\left(u \right)} - \int{\frac{u}{\sqrt{1 - u^{2}}} d u}\right)}}}{5}$$

Låt $$$v=1 - u^{2}$$$ vara.

Då $$$dv=\left(1 - u^{2}\right)^{\prime }du = - 2 u du$$$ (stegen kan ses »), och vi har att $$$u du = - \frac{dv}{2}$$$.

Integralen kan omskrivas som

$$- \frac{6 u \operatorname{asin}{\left(u \right)}}{5} + \frac{6 {\color{red}{\int{\frac{u}{\sqrt{1 - u^{2}}} d u}}}}{5} = - \frac{6 u \operatorname{asin}{\left(u \right)}}{5} + \frac{6 {\color{red}{\int{\left(- \frac{1}{2 \sqrt{v}}\right)d v}}}}{5}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ med $$$c=- \frac{1}{2}$$$ och $$$f{\left(v \right)} = \frac{1}{\sqrt{v}}$$$:

$$- \frac{6 u \operatorname{asin}{\left(u \right)}}{5} + \frac{6 {\color{red}{\int{\left(- \frac{1}{2 \sqrt{v}}\right)d v}}}}{5} = - \frac{6 u \operatorname{asin}{\left(u \right)}}{5} + \frac{6 {\color{red}{\left(- \frac{\int{\frac{1}{\sqrt{v}} d v}}{2}\right)}}}{5}$$

Tillämpa potensregeln $$$\int v^{n}\, dv = \frac{v^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=- \frac{1}{2}$$$:

$$- \frac{6 u \operatorname{asin}{\left(u \right)}}{5} - \frac{3 {\color{red}{\int{\frac{1}{\sqrt{v}} d v}}}}{5}=- \frac{6 u \operatorname{asin}{\left(u \right)}}{5} - \frac{3 {\color{red}{\int{v^{- \frac{1}{2}} d v}}}}{5}=- \frac{6 u \operatorname{asin}{\left(u \right)}}{5} - \frac{3 {\color{red}{\frac{v^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{5}=- \frac{6 u \operatorname{asin}{\left(u \right)}}{5} - \frac{3 {\color{red}{\left(2 v^{\frac{1}{2}}\right)}}}{5}=- \frac{6 u \operatorname{asin}{\left(u \right)}}{5} - \frac{3 {\color{red}{\left(2 \sqrt{v}\right)}}}{5}$$

Kom ihåg att $$$v=1 - u^{2}$$$:

$$- \frac{6 u \operatorname{asin}{\left(u \right)}}{5} - \frac{6 \sqrt{{\color{red}{v}}}}{5} = - \frac{6 u \operatorname{asin}{\left(u \right)}}{5} - \frac{6 \sqrt{{\color{red}{\left(1 - u^{2}\right)}}}}{5}$$

Kom ihåg att $$$u=5 x$$$:

$$- \frac{6 \sqrt{1 - {\color{red}{u}}^{2}}}{5} - \frac{6 {\color{red}{u}} \operatorname{asin}{\left({\color{red}{u}} \right)}}{5} = - \frac{6 \sqrt{1 - {\color{red}{\left(5 x\right)}}^{2}}}{5} - \frac{6 {\color{red}{\left(5 x\right)}} \operatorname{asin}{\left({\color{red}{\left(5 x\right)}} \right)}}{5}$$

Alltså,

$$\int{\left(- 6 \operatorname{asin}{\left(5 x \right)}\right)d x} = - 6 x \operatorname{asin}{\left(5 x \right)} - \frac{6 \sqrt{1 - 25 x^{2}}}{5}$$

Lägg till integrationskonstanten:

$$\int{\left(- 6 \operatorname{asin}{\left(5 x \right)}\right)d x} = - 6 x \operatorname{asin}{\left(5 x \right)} - \frac{6 \sqrt{1 - 25 x^{2}}}{5}+C$$

$$$\int \left(- 6 \operatorname{asin}{\left(5 x \right)}\right)\, dx = \left(- 6 x \operatorname{asin}{\left(5 x \right)} - \frac{6 \sqrt{1 - 25 x^{2}}}{5}\right) + C$$$A