Answered: "Use a table of integrals to evaluate the..."

quiquenobi2v6

Answered question

2021-12-12

Use a table of integrals to evaluate the following integrals.

\int x {(2 x + 3)}^{5} d x

boronganfh

Beginner2021-12-13Added 33 answers

Step 1
Consider the following integral,

I = \int x {(2 x + 3)}^{5} d x

use the substitution method of integral,
2x+3=u

d x = \frac{1}{2} d u

so, the integral becomes,

I = \frac{1}{4} \int (u - 3) u^{5} d u

now, simply the integral as follows,

I = \frac{1}{4} \int (u^{6} - 3 u^{5}) d u

= \frac{1}{4} \int u^{6} d u - \frac{3}{4} \int u^{5} d u

Step 2
now, integrate the above integral,

I = \frac{1}{4} [\frac{u^{7}}{7}] - \frac{3}{4} [\frac{u^{6}}{6}] + C

= \frac{1}{4} [\frac{{(2 x + 3)}^{7}}{7}] - \frac{3}{4} [\frac{{(2 x + 3)}^{6}}{6}] + C

= \frac{{(2 x + 3)}^{6} (4 x - 1)}{56} + C

so, the value of the integral is

I = \frac{{(2 x + 3)}^{6} (4 x - 1)}{56} + C

censoratojk

Beginner2021-12-14Added 46 answers

\int x {(2 x + 3)}^{5} d x

Substitution

u = 2 x + 3 \Rightarrow \frac{d u}{d x} = 2 \Rightarrow d x = \frac{1}{2} d u :

= \frac{1}{4} \int (u - 3) u^{5} d u

Now we calculate:

\int (u - 3) u^{5} d u

We use the distributive property:

= \int (u^{6} - 3 u^{5}) d u

Lets

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