Use two or more substitutions to find the following integral. $$ \int \frac{\mathrm{dx} ext {. }}{\sqrt{9+\sqrt{6+\mathrm{x}}}} $$ (Hint: Begin with $$ \mathrm{u}=\sqrt{6+\mathrm{x}} $$.) Find du. du $$ =\frac{1}{2 \sqrt{\mathrm{x+6}}} \mathrm{dx} $$ (Type an exact answer, using radicals as needed.) Rewrite the given integral using this change of variables. $$ \int \frac{\mathrm{dx}}{\sqrt{9+\sqrt{6+\mathrm{x}}}}=\int \frac{2 \mathrm{u}}{\sqrt{9+\mathrm{u}}} $$ du (Type an exact answer, using radicals as needed.) Find the indefinite integral. $$ \int \frac{\mathrm{dx}}{\sqrt{9+\sqrt{6+\mathrm{x}}}}=\square $$ (Type an exact answer, using radicals as needed.)

Question

Use two or more substitutions to find the following integral. $$ \int \frac{\mathrm{dx} 	ext {. }}{\sqrt{9+\sqrt{6+\mathrm{x}}}} $$ (Hint: Begin with $$ \mathrm{u}=\sqrt{6+\mathrm{x}} $$.) Find du. du $$ =\frac{1}{2 \sqrt{\mathrm{x+6}}} \mathrm{dx} $$ (Type an exact answer, using radicals as needed.) Rewrite the given integral using this change of variables. $$ \int \frac{\mathrm{dx}}{\sqrt{9+\sqrt{6+\mathrm{x}}}}=\int \frac{2 \mathrm{u}}{\sqrt{9+\mathrm{u}}} $$ du (Type an exact answer, using radicals as needed.) Find the indefinite integral. $$ \int \frac{\mathrm{dx}}{\sqrt{9+\sqrt{6+\mathrm{x}}}}=\square $$ (Type an exact answer, using radicals as needed.)

Moran Bob · Accepted Answer

The integral simplifies to $$ \frac{4}{3} (9 + \sqrt{6+x})^{3/2} - 36(9 + \sqrt{6+x})^{1/2} + C $$.

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