Testo
Si voglia svolgere il seguente integrale:
Soluzione
Come primo passaggio si può riscrivere come segue:
Sfruttando l’integrazione per parti sul secondo addendo si ha:
![\arcsin x-\left[x \sqrt{1-x^{2}}-\int-\frac{x^{2}}{\sqrt{1-x^{2}}}\right]=](https://s0.wp.com/latex.php?latex=%5Carcsin+x-%5Cleft%5Bx+%5Csqrt%7B1-x%5E%7B2%7D%7D-%5Cint-%5Cfrac%7Bx%5E%7B2%7D%7D%7B%5Csqrt%7B1-x%5E%7B2%7D%7D%7D%5Cright%5D%3D&bg=ffffff&fg=1a1a1a&s=0 "\arcsin x-\left[x \sqrt{1-x^{2}}-\int-\frac{x^{2}}{\sqrt{1-x^{2}}}\right]=")
![\arcsin x-\left[x \sqrt{1-x^{2}}-\int \frac{-x^{2}+(1-1)}{\sqrt{1-x^{2}}}\right]=](https://s0.wp.com/latex.php?latex=%5Carcsin+x-%5Cleft%5Bx+%5Csqrt%7B1-x%5E%7B2%7D%7D-%5Cint+%5Cfrac%7B-x%5E%7B2%7D%2B%281-1%29%7D%7B%5Csqrt%7B1-x%5E%7B2%7D%7D%7D%5Cright%5D%3D&bg=ffffff&fg=1a1a1a&s=0 "\arcsin x-\left[x \sqrt{1-x^{2}}-\int \frac{-x^{2}+(1-1)}{\sqrt{1-x^{2}}}\right]=")
![\arcsin x-\left[x \sqrt{1-x^{2}}-\int \frac{1-x^{2}-1}{\sqrt{1-x^{2}}}\right]=](https://s0.wp.com/latex.php?latex=%5Carcsin+x-%5Cleft%5Bx+%5Csqrt%7B1-x%5E%7B2%7D%7D-%5Cint+%5Cfrac%7B1-x%5E%7B2%7D-1%7D%7B%5Csqrt%7B1-x%5E%7B2%7D%7D%7D%5Cright%5D%3D&bg=ffffff&fg=1a1a1a&s=0 "\arcsin x-\left[x \sqrt{1-x^{2}}-\int \frac{1-x^{2}-1}{\sqrt{1-x^{2}}}\right]=")
![\arcsin x-\left[x \sqrt{1-x^{2}}-\left(\int \frac{1-x^{2}}{\sqrt{1-x^{2}}}-\int \frac{1}{\sqrt{1-x^{2}}}\right)\right]=](https://s0.wp.com/latex.php?latex=%5Carcsin+x-%5Cleft%5Bx+%5Csqrt%7B1-x%5E%7B2%7D%7D-%5Cleft%28%5Cint+%5Cfrac%7B1-x%5E%7B2%7D%7D%7B%5Csqrt%7B1-x%5E%7B2%7D%7D%7D-%5Cint+%5Cfrac%7B1%7D%7B%5Csqrt%7B1-x%5E%7B2%7D%7D%7D%5Cright%29%5Cright%5D%3D&bg=ffffff&fg=1a1a1a&s=0 "\arcsin x-\left[x \sqrt{1-x^{2}}-\left(\int \frac{1-x^{2}}{\sqrt{1-x^{2}}}-\int \frac{1}{\sqrt{1-x^{2}}}\right)\right]=")
![\arcsin x-\left[x \sqrt{1-x^{2}}-\left(\int \frac{1-x^{2}}{\sqrt{1-x^{2}}} \cdot \frac{\sqrt{1-x^{2}}}{\sqrt{1-x^{2}}}-\int \frac{1}{\sqrt{1-x^{2}}}\right)\right]=](https://s0.wp.com/latex.php?latex=%5Carcsin+x-%5Cleft%5Bx+%5Csqrt%7B1-x%5E%7B2%7D%7D-%5Cleft%28%5Cint+%5Cfrac%7B1-x%5E%7B2%7D%7D%7B%5Csqrt%7B1-x%5E%7B2%7D%7D%7D+%5Ccdot+%5Cfrac%7B%5Csqrt%7B1-x%5E%7B2%7D%7D%7D%7B%5Csqrt%7B1-x%5E%7B2%7D%7D%7D-%5Cint+%5Cfrac%7B1%7D%7B%5Csqrt%7B1-x%5E%7B2%7D%7D%7D%5Cright%29%5Cright%5D%3D+&bg=ffffff&fg=1a1a1a&s=0 "\arcsin x-\left[x \sqrt{1-x^{2}}-\left(\int \frac{1-x^{2}}{\sqrt{1-x^{2}}} \cdot \frac{\sqrt{1-x^{2}}}{\sqrt{1-x^{2}}}-\int \frac{1}{\sqrt{1-x^{2}}}\right)\right]=")
![\arcsin x-\left[x \sqrt{1-x^{2}}-\int \sqrt{1-x^{2}}+\int \frac{1}{\sqrt{1-x^{2}}}\right]=](https://s0.wp.com/latex.php?latex=%5Carcsin+x-%5Cleft%5Bx+%5Csqrt%7B1-x%5E%7B2%7D%7D-%5Cint+%5Csqrt%7B1-x%5E%7B2%7D%7D%2B%5Cint+%5Cfrac%7B1%7D%7B%5Csqrt%7B1-x%5E%7B2%7D%7D%7D%5Cright%5D%3D+&bg=ffffff&fg=1a1a1a&s=0 "\arcsin x-\left[x \sqrt{1-x^{2}}-\int \sqrt{1-x^{2}}+\int \frac{1}{\sqrt{1-x^{2}}}\right]=")
Per la quantità tra parentesi quadre si nota che:
Quindi:
Ovvero:
Quindi tutta la quantità tra parentesi quadre può essere sostituita con quella appena calcolata:
![\arcsin x-\left[\frac{x \sqrt{1-x^{2}}}{2}+\frac{\arcsin x}{2}\right]+c](https://s0.wp.com/latex.php?latex=%5Carcsin+x-%5Cleft%5B%5Cfrac%7Bx+%5Csqrt%7B1-x%5E%7B2%7D%7D%7D%7B2%7D%2B%5Cfrac%7B%5Carcsin+x%7D%7B2%7D%5Cright%5D%2Bc&bg=ffffff&fg=1a1a1a&s=0 "\arcsin x-\left[\frac{x \sqrt{1-x^{2}}}{2}+\frac{\arcsin x}{2}\right]+c")
Il termine costante definisce le altre primitive.
Al finale, sommando, si ha che:
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