Calculus for Team-Based Inquiry Learning: 2022 Edition

Activity 5.3.1.

Consider ${\displaystyle \int \sin (x)\cos (x)dx\text{.}}$ Which strategy would you choose to evaluate this integral?

1. The method of substitution

2. Integration by parts

3. Other (specify your strategy)

Activity 5.3.2.

Consider ${\displaystyle \int {\sin }^2(x)\cos (x)dx\text{.}}$ Which strategy would you choose to evaluate this integral?

1. The method of substitution

2. Integration by parts

3. Other (specify your strategy)

Activity 5.3.3.

Consider ${\displaystyle \int {\sin }^2(x){\cos }^2(x)dx\text{.}}$ Which strategy would you choose to evaluate this integral?

1. The method of substitution

2. Integration by parts

3. Other (specify your strategy)

Activity 5.3.4.

Activity 5.3.5.

Consider ${\displaystyle \int {\sin }^m(x){\cos }^n(x)dx\text{,}}$ where $m$ is even and $n$ is odd. Which strategy would you choose to evaluate this integral?

1. Rewrite sine in terms of cosine by using a double-angle/half-angle identity.

2. Rewrite cosine in terms of sine by using a double-angle/half-angle identity.

3. Rewrite cosine in terms of sine by using a pythagorean identity.

Activity 5.3.6.

Consider ${\displaystyle \int {\sin }^m(x){\cos }^n(x)dx\text{,}}$ where $m$ and $n$ are both odd and $m\ge n\text{.}$ Which strategy would you choose to evaluate this integral?

1. Rewrite sine in terms of cosine by using a pythagorean identity.

2. Rewrite cosine in terms of sine by using a double-angle/half-angle identity.

3. Rewrite cosine in terms of sine by using a pythagorean identity.

Activity 5.3.7.

Consider ${\displaystyle \int {\sin }^m(x){\cos }^n(x)dx\text{,}}$ where $m$ and $n$ are both even and $m\ge n\text{.}$ Which strategy would you choose to evaluate this integral?

1. Rewrite sine in terms of cosine by using a pythagorean identity.

2. Rewrite cosine in terms of sine by using a pythagorean identity.

3. Rewrite everything in terms of cosine by using a double-angle/half-angle identity.

Activity 5.3.8.

Consider ${\displaystyle \int {\sin }^4(x){\cos }^4(x)dx\text{.}}$ Which would be the most useful way to rewrite the integral?

1. ${\displaystyle {\displaystyle \int (1-{\cos }^2(x){)}^2{\cos }^4(x)dx}}$

2. ${\displaystyle {\displaystyle \int {\sin }^4(x)(1-{\sin }^2(x){)}^{2}dx}}$

3. ${\displaystyle {\displaystyle \int {\left(\frac{1-\cos (2x)}{2}\right)}^2{\left(\frac{1+\cos (2x)}{2}\right)}^{2}dx}}$

Activity 5.3.9.

Consider ${\displaystyle \int {\sin }^3(x){\cos }^5(x)dx\text{.}}$ Which would be the most useful way to rewrite the integral?

1. ${\displaystyle {\displaystyle \int (1-{\cos }^2(x))\sin (x){\cos }^5(x)dx}}$

2. ${\displaystyle {\displaystyle \int {\sin }^3(x){\left(\frac{1+\cos (2x)}{2}\right)}^2\cos (x)dx}}$

3. ${\displaystyle {\displaystyle \int {\sin }^3(x)(1-{\sin }^2(x){)}^2\cos (x)dx}}$

Activity 5.3.11.

Consider ${\displaystyle \int \sin (\theta )\sin (3\theta )d\theta \text{.}}$

1. Find an identity from Appendix B which could be used to transform our integrand.

2. Rewrite the integrand using the selected identity.

3. Evaluate the integral.