Calc 7.0 Mental u-sub Exploration

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​7.0 Mental u-Sub

By Elizabeth Mendenhall

Caution!

Caveat:  This works if u is a first-degree polynomial
(because du = constant * dx). 

With powers greater than 1, it’s worth the time to write out the u-substitution.

Fill in the Blanks

You should use mental u-sub when $u=x^?$

has a power of _.

Multiple Choice

Explore, using regular u-sub. Is there a pattern/shortcut?

$$\int_{ }^{ }\cos\left(2x\right)dx\ =\ ?$$

• $$-\frac{1}{2}\sin\left(2x\right)+C$$

• $$-2\sin\left(2x\right)+C$$

• $$\frac{1}{2}\sin\left(2x\right)+C$$

• $$2\sin\left(2x\right)+C$$

Multiple Choice

Explore, using regular u-sub. Is there a pattern/shortcut?

$$\int_{ }^{ }\cos\left(\frac{1}{4}x\right)dx\ =\ ?$$

• $$\frac{1}{4}\sin\left(\frac{1}{4}x\right)+C$$

• $$4\sin\left(\frac{1}{4}x\right)+C$$

• $$-\frac{1}{4}\sin\left(\frac{1}{4}x\right)+C$$

• $$-4\sin\left(\frac{1}{4}x\right)+C$$

​In u-sub, when x has a coefficient, what pattern did you observe?

Multiple Choice

How often does this pattern work?

$$\int_{ }^{ }\cos\left(2x-1\right)dx\ =\ ?$$

HINT: u = 2x -1 → du = 2 dx → constant is 1/2

• $$\frac{1}{2}\sin\left(2x-1\right)+C$$

• $$2\sin\left(2x-1\right)+C$$

Multiple Choice

What if there's an e?

$$\int_{ }^{ }e^{-2x}dx\ =\ ?$$

HINT: u = -2x → du = -2 dx → constant is -1/2

• $$-2e^{-2x}+C$$

• $$-\frac{1}{2}e^{-2x}+C$$

Use mental u-sub for each. How?
1. Find u.
2. Then find and flip du.

​

​Practice time!

Multiple Choice

$$\int_{ }^{ }\sin\left(2x\right)\ dx\ =$$

• $$-\frac{1}{2}\cos\left(2x\right)+C$$

• $$\frac{1}{2}\cos\left(2x\right)+C$$

• $$-2\cos\left(2x\right)+C$$

• $$2\cos\left(2x\right)+C$$

Multiple Choice

$$\int_{ }^{ }e^{3x}\ dx\ =$$

• $$3e^{3x}+C$$

• $$\frac{1}{3}e^{3x}+C$$

• $$\frac{1}{3x}e^{3x}+C$$

• $$3xe^{3x}+C$$

Multiple Choice

Quick review before you try this with u-sub: $$\int_{ }^{ }\frac{1}{x}\ dx\ =$$

• $$\ln\left|x\right|+C$$

• $$\frac{x^0}{0}+C$$

• $$-\frac{1}{x^2}+C$$

Fill in the Blanks

$\int_{ }^{ }\frac{1}{1+2x}\ dx$

For this integral, what is u?

Multiple Choice

Now do this carefully:

$$\int_{ }^{ }\frac{1}{1+2x}dx\ =$$

• $$\frac{1}{2}\ln\left|1+2x\right|+C$$

• $$2\ln\left|1+2x\right|+C$$

• $$\ln\left|1+2x\right|+C$$

Multiple Choice

$$\int_{ }^{ }\frac{1}{2+x}dx\ =$$

• $$\frac{1}{2}\ln\left|2+x\right|+C$$

• $$2\ln\left|2+x\right|+C$$

• $$\ln\left|2+x\right|+C$$

Multiple Choice

$$\int_{ }^{ }\frac{1}{2+7x}dx\ =$$

• $$\frac{1}{7}\ln\left|2+7x\right|+C$$

• $$7\ln\left|2+7x\right|+C$$

• $$\frac{2}{7}\ln\left|2+7x\right|+C$$

Multiple Choice

$$\int_{ }^{ }\frac{7}{2+7x}dx\ =$$

• $$\frac{1}{7}\ln\left|2+7x\right|+C$$

• $$7\ln\left|2+7x\right|+C$$

• $$\ln\left|2+7x\right|+C$$

Multiple Choice

$$\int_{ }^{ }-4\sin\left(\frac{1}{2}x\right)\ dx\ =$$

• $$-2\cos\left(\frac{1}{2}x\right)+C$$

• $$2\cos\left(\frac{1}{2}x\right)+C$$

• $$-8\cos\left(\frac{1}{2}x\right)+C$$

• $$8\cos\left(\frac{1}{2}x\right)+C$$

Multiple Choice

$$\int_{ }^{ }2e^{3x}\ dx$$  

• $$6e^{3x}+c$$  

• $$\frac{2}{3}e^{3x}+c$$  

• $$2e^{\frac{3}{2}x^2}+c$$  

• $$2e^{\frac{3}{2}x}+c$$  

Multiple Choice

$$\int_{ }^{ }e^{6-11x}\ dx$$

• $$-11e^{\left(6x-\frac{11}{2}x^2\right)}+C$$

• $$-11e^{6-11x}+C$$

• $$-\frac{1}{11}e^{6-11x}+C$$

• $$-\frac{1}{11}e^{\left(6x-\frac{11}{2}x^2\right)}+C$$

Multiple Choice

$$\int\ \frac{1}{2}e^{\frac{1}{2}x}\ dx$$

• $$\frac{1}{4}e^{\frac{1}{2}x}+C$$

• $$e^{\frac{1}{2}x}+C$$

• $$\frac{1}{2}e^{\frac{1}{2}x}+C$$

Multiple Choice

Reminder: if x has a degree greater than 1, use traditional u-sub.

$$\int x\cos(x^2)dx$$

• $$-2\sin\left(x^2\right)+C$$

• $$2\sin\left(x^2\right)+C$$

• $$-\frac{1}{2}\sin\left(x^2\right)+C$$

• $$\frac{1}{2}\sin\left(x^2\right)+C$$

Multiple Choice

Reminder: if x has a degree greater than 1, use traditional u-sub.

$$\int\cos\left(4x\right)\ e^{\sin\left(4x\right)}dx$$

• $$4e^{\sin\left(4x\right)}+C$$

• $$\frac{1}{4}e^{\sin\left(4x\right)}+C$$

• $$-4e^{\sin\left(4x\right)}+C$$

• $$-\frac{1}{4}e^{\sin\left(4x\right)}+C$$

Multiple Choice

Write down as many steps as you need: $$\int_1^2\frac{1}{2x+1}dx$$

• $$2\ \ln\ 2$$

• $$\frac{1}{2}\ln2$$

• $$2\left(\ln5-\ln3\right)$$

• $$\frac{1}{2}\left(\ln5-\ln3\right)$$

Poll

How is mental u-sub going?

• I'm confident

• I have to be careful to not make mistakes

• I need more practice

• Help!