Integrating Using Substitution - AP Calculus AB

1. Determine $du$ if $u = \text{ln } x$ in substitution.

   $du = \frac{1}{x} \, dx$. The derivative of $u = \ln x$ is $\frac{1}{x}$, so $du = \frac{1}{x} dx$.

2. Determine $du$ if $u = \text{ln}(4x)$ in substitution.

   $du = \frac{1}{x} \, dx$. The derivative of $u = \ln(4x)$ is $\frac{1}{x}$, so $du = \frac{1}{x} dx$.

3. Identify $u$ for substitution if $du = \text{sec}^2 x \, dx$.

   $u = \text{tan } x$. Since $\frac{d}{dx}(\tan x) = \sec^2 x$, if $du = \sec^2 x dx$, then $u = \tan x$.

4. What substitution simplifies the integral of $(2x + 1)^5$?

   $u = 2x + 1$. Let $u = 2x + 1$ to transform $(2x + 1)^5$ into $u^5$ for easier integration.

5. What is the integral of $e^u \, du$?

   $e^u + C$. The antiderivative of $e^u$ is $e^u$ plus the constant of integration.

6. Determine $du$ if $u = 2x^3 + 1$ in substitution.

   $du = 6x^2 \, dx$. The derivative of $u = 2x^3 + 1$ is $6x^2$, so $du = 6x^2 dx$.

7. Identify $du$ if $u = x^3$ in substitution.

   $du = 3x^2 \, dx$. The derivative of $u = x^3$ is $\frac{du}{dx} = 3x^2$, so $du = 3x^2 dx$.

8. What is the integral of $\frac{1}{u} \, du$?

   $\text{ln } |u| + C$. The antiderivative of $\frac{1}{u}$ is $\ln|u|$ plus the constant of integration.

9. Determine $du$ for $u = \text{sin } x$ in substitution.

   $du = \text{cos } x \, dx$. The derivative of $u = \sin x$ is $\cos x$, so $du = \cos x dx$.

10. Find $u$ for substitution if $du = 7x^6 \, dx$.

    $u = x^7$. Since $\frac{d}{dx}(x^7) = 7x^6$, if $du = 7x^6 dx$, then $u = x^7$.

11. Find the integral of $\text{sec}^2 u \, du$.

    $\text{tan } u + C$. The antiderivative of $\sec^2 u$ is $\tan u$ plus the constant of integration.

12. Find $u$ for substitution if $du = 3x^2 dx$ in $\frac{1}{(x^3+1)^2}$.

    $u = x^3 + 1$. If $du = 3x^2 dx$, then $u = x^3 + 1$ since $\frac{d}{dx}(x^3 + 1) = 3x^2$.

13. What substitution simplifies the integral of $x e^{x^2}$?

    $u = x^2$. Let $u = x^2$ so that $du = 2x dx$ matches the $x$ factor in $xe^{x^2}$.

14. Identify $u$ for substitution if $du = 5x^4 \, dx$.

    $u = x^5$. Since $\frac{d}{dx}(x^5) = 5x^4$, if $du = 5x^4 dx$, then $u = x^5$.

15. What is the integral of $\text{sin } u \, du$?

    $-\text{cos } u + C$. The antiderivative of $\sin u$ is $-\cos u$ plus the constant of integration.

16. Find $u$ for substitution if $du = \frac{1}{x} \, dx$.

    $u = \text{ln } x$. Since $\frac{d}{dx}(\ln x) = \frac{1}{x}$, if $du = \frac{1}{x} dx$, then $u = \ln x$.

17. Find $u$ for substitution if $du = \frac{1}{x^2} \, dx$.

    $u = -\frac{1}{x}$. Since $\frac{d}{dx}(-\frac{1}{x}) = \frac{1}{x^2}$, if $du = \frac{1}{x^2} dx$, then $u = -\frac{1}{x}$.

18. What substitution simplifies the integral of $\text{cos}(x^2)$?

    $u = x^2$. Let $u = x^2$ to simplify the argument of the cosine function.

19. Determine $du$ if $u = 3x^4 + 2x$ in substitution.

    $du = (12x^3 + 2) \, dx$. The derivative of $u = 3x^4 + 2x$ is $12x^3 + 2$, so $du = (12x^3 + 2) dx$.

20. What is the integral of $u^n \, du$?

    $\frac{u^{n+1}}{n+1} + C$, $n \neq -1$. Use the power rule for integration: increase exponent by 1 and divide.

21. What is the integral of $\text{cos } u \, du$?

    $\text{sin } u + C$. The antiderivative of $\cos u$ is $\sin u$ plus the constant of integration.

22. Determine $du$ if $u = x^2 - 1$ in substitution.

    $du = 2x \, dx$. The derivative of $u = x^2 - 1$ is $2x$, so $du = 2x dx$.

23. Identify $u$ for substitution if $du = 4x^3 \, dx$.

    $u = x^4$. Since $\frac{d}{dx}(x^4) = 4x^3$, if $du = 4x^3 dx$, then $u = x^4$.

24. Determine $du$ if $u = 5x + 3$ in substitution.

    $du = 5 \, dx$. The derivative of $u = 5x + 3$ is $5$, so $du = 5 dx$.

25. What substitution simplifies the integral of $x \text{cos}(x^2)$?

    $u = x^2$. Let $u = x^2$ so that $du = 2x dx$ matches the $x$ factor in $x\cos(x^2)$.

26. Identify $u$ for substitution if $du = e^x \, dx$.

    $u = e^x$. Since $\frac{d}{dx}(e^x) = e^x$, if $du = e^x dx$, then $u = e^x$.

27. What is the substitution for $u$ if $du = -\text{sin } x \, dx$?

    $u = \text{cos } x$. Since $\frac{d}{dx}(\cos x) = -\sin x$, if $du = -\sin x dx$, then $u = \cos x$.

28. What substitution simplifies the integral of $\cos(x^2)$?

    $u = x^2$. Let $u = x^2$ to simplify the argument of the cosine function.

29. Identify $u$ for substitution if $du = 2x \, dx$.

    $u = x^2$. Since $\frac{d}{dx}(x^2) = 2x$, if $du = 2x dx$, then $u = x^2$.

30. Find $u$ for substitution if $du = \text{cos } x \, dx$.

    $u = \text{sin } x$. Since $\frac{d}{dx}(\sin x) = \cos x$, if $du = \cos x dx$, then $u = \sin x$.

31. Determine $du$ if $u = \text{tan } x$ in substitution.

    $du = \text{sec}^2 x \, dx$. The derivative of $u = \tan x$ is $\sec^2 x$, so $du = \sec^2 x dx$.

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