Advanced Substitution

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Evaluate the indefinite integral $\int x^5 \sqrt{1 - x^2} \, dx$ .

-\frac{1}{3}(1-x^2)^{3/2} + \frac{2}{5}(1-x^2)^{5/2} - \frac{1}{7}(1-x^2)^{7/2} + C

\frac{1}{3}(1-x^2)^{3/2} - \frac{2}{5}(1-x^2)^{5/2} + \frac{1}{7}(1-x^2)^{7/2} + C

-\frac{2}{3}(1-x^2)^{3/2} + \frac{4}{5}(1-x^2)^{5/2} - \frac{2}{7}(1-x^2)^{7/2} + C

-\frac{1}{3}(1-x^2)^{3/2} + \frac{1}{5}(1-x^2)^{5/2} + C

Evaluate $\int \frac{dx}{\sqrt{2x - x^2}}$ .

\arcsin(2x-1) + C

\arctan(x-1) + C

\arcsin(x) + C

\arcsin(x-1) + C

Evaluate the integral $\int \frac{\sqrt{1+\sqrt{x}}}{\sqrt{x}} \, dx$ .

\frac{2}{3} (1+\sqrt{x})^{3/2} + C

\frac{4}{3} (1+\sqrt{x})^{3/2} + C

\frac{4}{3} (1+\sqrt{x})^{3/2} \cdot \frac{1}{\sqrt{x}} + C

\frac{1}{3} (1+\sqrt{x})^{3/2} + C

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