Selecting Techniques for Antidifferentiation - AP Calculus BC

1. Which technique is most appropriate for $\int x\ln(x)\,dx$?

   Use integration by parts. Products of polynomials and logarithms require parts.

2. What is the antiderivative of $e^{ax}$?

   $\frac{1}{a}e^{ax} + C$. Chain rule in reverse: derivative of $\frac{1}{a}e^{ax}$ is $e^{ax}$.

3. Which technique is most appropriate for $\int e^{3x}\,dx$?

   Use reverse chain rule: $\int e^{ax}dx=\frac{1}{a}e^{ax}+C$. The coefficient $a$ in $e^{ax}$ moves to the denominator.

4. Which technique is most appropriate for $\int e^x\cos(x)\,dx$?

   Use integration by parts twice (tabular/recursive). Products of exponentials and trig functions need repeated parts.

5. Which technique is most appropriate for $\int x\sqrt{x^2+4}\,dx$?

   Use $u$-substitution with $u=x^2+4$. The $x$ factor is half the derivative of what's under the radical.

6. Which technique is most appropriate for $\int \sqrt{x^2+16}\,dx$?

   Use trig substitution: $x=4\tan(\theta)$. Square root of $x^2+a^2$ form requires tangent substitution.

7. Which technique is most appropriate for $\int \sqrt{x^2-16}\,dx$?

   Use trig substitution: $x=4\sec(\theta)$. Square root of $x^2-a^2$ form requires secant substitution.

8. Which technique is most appropriate for $\int \frac{1}{\sqrt{9-x^2}}\,dx$?

   Use arcsine form: $\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin(\frac{x}{a})+C$. Matches the arcsine integral form with $a=3$.

9. Which technique is most appropriate for $\int x e^{x^2}\,dx$?

   Use $u$-substitution with $u=x^2$. The $x$ factor is half the derivative of the exponent.

10. Which technique is used for $xe^{x^2}$?

    U-Substitution. Let $u = x^2$, then $du = 2x dx$ for substitution.

11. Which technique is most appropriate for $\int \frac{x}{x^2+9}\,dx$?

    Use $u$-substitution with $u=x^2+9$. The numerator $x$ is half the derivative of the denominator.

12. Which technique is most appropriate for $\int \frac{1}{x^2-9}\,dx$?

    Use partial fractions (factor $x^2-9=(x-3)(x+3)$). Factorable quadratic denominator uses partial fractions.

13. Which technique is most appropriate for $\int \frac{1}{x^2+9}\,dx$?

    Use arctangent form: $\int \frac{1}{x^2+a^2}dx=\frac{1}{a}\arctan(\frac{x}{a})+C$. Matches the arctangent integral form with $a=3$.

14. Which technique is most appropriate for $\int \sqrt{9-x^2}\,dx$?

    Use trig substitution: $x=3\sin(\theta)$. Square root of $a^2-x^2$ form requires sine substitution.

15. What is the antiderivative of $x^n$ for $n \neq -1$?

    $\frac{x^{n+1}}{n+1} + C$. Power rule for integration: increase exponent by 1, divide by new exponent.

16. Which technique is most appropriate for $\int x\sin(x)\,dx$?

    Use integration by parts. Products of polynomials and trig functions require parts.

17. Which technique is most appropriate for $\int \sin^3(x)\cos^2(x)\,dx$?

    Use trig identities; save a $\sin(x)$ and substitute $u=\cos(x)$. Odd power of sine allows $u$-substitution after saving one sine.

18. Which method is used for $x^2 e^x$?

    Integration by Parts. Product $x^2 e^x$ requires repeated integration by parts.

19. Which technique is most appropriate for $\int (x^2+1)^7\cdot 2x\,dx$?

    Use $u$-substitution with $u=x^2+1$. The derivative of $u=x^2+1$ is $2x$, which appears in the integrand.

20. Which technique is most appropriate for $\int \frac{1}{2x-5}\,dx$?

    Use reverse chain rule: $\frac{1}{2}\ln|2x-5|+C$. Linear denominator requires chain rule adjustment by $\frac{1}{2}$.

21. Which technique is most appropriate for $\int \cos(7x)\,dx$?

    Use reverse chain rule: $\int \cos(ax)dx=\frac{1}{a}\sin(ax)+C$. The coefficient $a$ in $\cos(ax)$ moves to the denominator.

22. Which technique is most appropriate for $\int \frac{1}{x}\,dx$?

    Use the logarithm rule: $\int \frac{1}{x}dx=\ln|x|+C$. The integral of $\frac{1}{x}$ is the natural logarithm.

23. What is the antiderivative of $e^x$?

    $e^x + C$. The derivative of $e^x$ is itself, so integration reverses this.

24. Which technique is most appropriate for $\int \frac{x^2+1}{x-3}\,dx$?

    Use algebraic division, then integrate term-by-term. Improper rational function requires polynomial division first.

25. Which technique is most appropriate for $\int x^5\,dx$?

    Use the power rule (basic antiderivative). Polynomial integrands use the power rule directly.

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