"Right, or wrong? Say which for each formula..." was answered by our experts

Caelan

Answered question

2021-02-20

Right, or wrong? Say which for each formula and give a brief reason for each answer.

\int (2 x + 1) 2 d x = (2 x + 1) \frac{3}{3} + C

hesgidiauE

Skilled2021-02-21Added 106 answers

Step 1
Consider the integral,

\int {(2 x + 1)}^{2} d x

Step 2
To solve the given integrals,

\int {(2 x + 1)}^{2} d x = \int (4 x^{2} + 1 + 4 x) d x

= 4 \int x^{2} d x + \int 1 \cdot d x + 4 \int x d x

= 4 (\frac{x^{3}}{3}) + x + 4 (\frac{x^{2}}{2}) + C

= \frac{4 x^{3}}{3} + 2 x^{2} + x + C

Step 3

\int {(2 x + 1)}^{2} d x = \frac{{(2 x + 1)}^{3}}{3 \cdot 2} + C

= \frac{8 x^{3} + 1 + 6 x (2 x + 1)}{6} + C

= \frac{8 x^{3} + 1 + 12 x^{2} + 6 x}{6} + C

= \frac{8 x^{3}}{6} + \frac{12 x^{2}}{6} + 6 \frac{x}{6} + \frac{1}{6} + C

= \frac{4 x^{3}}{3} + 2 x^{2} + x + \frac{1}{6} + C

Hence the given integral:

\int {(2 x + 1)}^{2} d x = \frac{{(2 x + 1)}^{3}}{3} + C

is wrong.
The right answer is:

\int {(2 x + 1)}^{2} d x = \frac{{(2 x + 1)}^{3}}{6} + C

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