Resolved: "Use a substitution of the form u =..."

leviattan0pi

Answered question

2021-11-16

Use a substitution of the form u = ax + b to evaluate the following indefinite integrals.

\int (x + 1) 12 d x

Ruth Phillips

Beginner2021-11-17Added 18 answers

Step 1
Given:

\int (x + 1) 12 d x

for evaluating this integral we substitute
u=x+1
now differentiate both side with respect to u

1 = \frac{d x}{d u} + 0

1 = \frac{d x}{d u}

du=dx
now replacing dx with du in given integral and integrate it
Step 2
so,

\int 12 u d u = 12 \int u d u (∵ \int x d x = \frac{x^{2}}{2} + c)

= 12 (\frac{u^{2}}{2}) + c

= 6 u^{2} + c

now putting u=x+1
so,

\int 12 (x + 1) d x = 6 {(x + 1)}^{2} + c

hence, given integral is equal to

6 {(x + 1)}^{2} + c

Zachary Pickett

Beginner2021-11-18Added 17 answers

Step 1: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

12 \int x + 1 d x

Step 2: Use Power Rule:

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C

12 (\frac{x^{2}}{2} + x)

Step 3: Add constant.

12 (\frac{x^{2}}{2} + x) + C

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?