1. $\int \frac{x^3}{\sqrt{9 - x^2}} \, dx = -\frac{1}{3}(9 - x^2)^{3/2} + C$
2. $\int \frac{1}{x^2\sqrt{4 - x^2}} \, dx = -\frac{\sqrt{4 - x^2}}{4x} + C$
3. $\int x^2\sqrt{a^2 - x^2} \, dx = \frac{1}{8}a^4\left(\arcsin\left(\frac{x}{a}\right) - \frac{x}{a}\sqrt{1 - \frac{x^2}{a^2}}\right) + C$
4. $\int x\sqrt{1 - x^2} \, dx = -\frac{1}{3}(1 - x^2)^{3/2} + C$
5. $\int \sqrt{4 - 9x^2} \, dx = \frac{1}{2}x\sqrt{4 - 9x^2} + \frac{2}{3}\arcsin\left(\frac{3x}{2}\right) + C$
6. $\int \frac{x^3}{\sqrt{9 - x^2}} \, dx = -\frac{1}{3}(9 - x^2)^{3/2} + C$
7. $\int \sqrt{x^2 - 4} \, dx = \frac{1}{2}x\sqrt{x^2 - 4} - 2\ln\left|x + \sqrt{x^2 - 4}\right| + C$
8. $\int \frac{x^3}{(3 + 4x - x^2)^{3/2}} \, dx = -\frac{1}{\sqrt{3 + 4x - x^2}} + C$
9. $\int \frac{\sqrt{25 - x^2}}{x} \, dx = \sqrt{25 - x^2} - 5\ln\left|\frac{5 + \sqrt{25 - x^2}}{x}\right| + C$
10. $\int \frac{\sqrt{x^2 - 4}}{x} \, dx = \sqrt{x^2 - 4} - 2\arctan\left(\frac{\sqrt{x^2 - 4}}{2}\right) + C$
11. $\int \frac{\sqrt{x^2 - 1}}{x^2} \, dx = \ln\left|x + \sqrt{x^2 - 1}\right| - \frac{\sqrt{x^2 - 1}}{x} + C$
12. $\int \frac{\sqrt{x^2 - 9}}{x^2} \, dx = \ln\left|x + \sqrt{x^2 - 9}\right| - \frac{\sqrt{x^2 - 9}}{x} + C$
13. $\int \frac{x^3}{\sqrt{x^2 - 16}} \, dx = \frac{1}{3}(x^2 - 16)^{3/2} + 16\sqrt{x^2 - 16} + C$
14. $\int_1^2 \frac{1}{(x^2 - 1)^{3/2}} \, dx = \frac{2}{\sqrt{3}}$
15. $\int \frac{\sqrt{3 + x^2}}{x} \, dx = \sqrt{3 + x^2} - \sqrt{3}\ln\left|\frac{\sqrt{3 + x^2} + \sqrt{3}}{x}\right| + C$
16. $\int \frac{dx}{\sqrt{x^2 + 2x + 5}} = \ln\left|x + 1 + \sqrt{x^2 + 2x + 5}\right| + C$
17. $\int \frac{dx}{x^2\sqrt{x^2 + 4}} = -\frac{\sqrt{x^2 + 4}}{4x} + C$
18. $\int \frac{dx}{\sqrt{x^2 - 6x + 13}} = \ln\left|x - 3 + \sqrt{x^2 - 6x + 13}\right| + C$
19. $\int \frac{dx}{\sqrt{1 + x^2}} = \ln\left|x + \sqrt{1 + x^2}\right| + C$
20. $\int_0^1 \frac{1}{(x^2 + 1)^2} \, dx = \frac{\pi}{8} + \frac{1}{4}$
Parent Tip: Review the logic above to help your child master the concept of integration by substitution worksheet.
