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Aus Online Mathematik Brückenkurs 2

(HINT: multiply the top and bottom by the conjugate of the denominator.)

If we multiply top and bottom of the fraction by the conjugate expression, x+9+x , then the conjugate rule gives that denominator's root is squared away:

1x+9−x=1x+9−xx+9+xx+9+x=x+9+xx+92−x2=x+9−xx+9+x=9x+9+x

Thus,

dxx+9−x=91x+9+xdx 

If we write the square roots in power form,

91x+921+x21dx ,

we see that we have a standard integral and can write down the primitive functions directly:

91x+921+x21dx=9121+1x+921+1+x21+121+1+C=9123x+923+23x23+C=9132x+923+32x23+C=227x+923+227x23+C,

where C is an arbitrary constant.

This can also be written with square roots as

227x+9x+9+227xx+C 

To be completely certain that we have everything correctly, we differentiate the answer and see if we get back the integrand:

ddx227x+923+227x23+C=22723x+923−1+22723x23−1+0=91x+921+91x21