PRAXIS: Antiderivatives

Compute each of the following indefinite integrals. Note that some algebra might be needed to get things into a form that can be easy antiderived. DON’T FORGET “+C” ! Also, remember: no work shown (where possible), no points!

1. \(\int x^2+2x+1\ dx\)
2. \(\int 4x^5-3x^4+2x+5\ dx\)
3. \(\int 0.124x^3-0.02x^2+x-1.25\ dx\)
4. \(\int \sqrt{x}+\sqrt[3]{x}-2\sqrt[5]{x}\ dx\)
5. \(\int \frac{1}{x}-2\frac{1}{x^2}+\frac{5}{x^3}\ dx\)
6. \(\int \frac{x^2+\sqrt{x}+3x}{x^2}\ dx\)

Find the antiderivative \(F\) for each of the following functions given the initial conditions provided. You may reuse calculations done above.

7. \(f(x)=x^2+2x+1\) where \(F(0)=5\)
8. \(f(x)=4x^5-3x^4+2x+5\) where \(F(1)=2\)
9. \(f(x)=0.5x^3-1.25x^2-0.25x+1\) where \(F(0.5)=1\)

Use the substitution method to compute the following indefinite integrals. Don’t forget “+C”!

10. \(\int 2x(x^2-5)^4\ dx\)
11. \(\int x^2( 5x^3-2)^2\ dx\)
12. \(\int (x^2+2x)\sqrt{x^3+3x^2}\ dx\)
13. \(\int x^3e^{x^4+1}\ dx\)
14. \(\int e^{x^5-3x^2+1}\cdot (x^4-\frac{6}{5}x)\ dx\)
15. \(\int\frac{ e^{\sqrt{x}+2}}{\sqrt{x}}\ dx\)
16. \(\int \frac{2x+1}{x^2+x+3}\ dx\)
17. \(\int \frac{x+1}{(x+1)^2}\ dx\)
18. \(\int \frac{\ln(x)}{x}\ dx\)