Explained: "Evaluate the indefinite integral. ∫(2x3−7)2dx"

Liesehf

Answered question

2021-11-17

Evaluate the indefinite integral.

\int {(2 x^{3} - 7)}^{2} d x

Tamara Donohue

Beginner2021-11-18Added 11 answers

Step 1
We have to evaluate

\int {(2 x^{3} - 7)}^{2} d x

We know that

{(a - b)}^{2} = a^{2} - 2 a b + b^{2}

And

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

Integrating the given integral using the above formula,

\int {(2 x^{3} - 7)}^{2} d x = \int [{(2 x^{3})}^{2} - 2 \cdot 2 x^{3} \cdot 7 + {(7)}^{2}] d x

= \int [4 x^{6} - 28 x^{3} + 49] d x

= \int 4 x^{6} d x - \int 28 x^{3} + \int 49 d x

= 4 \cdot \frac{x^{6 + 1}}{6 + 1} - 28 \cdot \frac{x^{3 + 1}}{3 + 1} + 49 x + C

, where C is integration constant.

= \frac{4}{7} x^{7} - 7 x^{4} + 49 x + C

Step 2
Hence, required integral is

\frac{4}{7} x^{7} - 7 x^{4} + 49 x + C

Nicole Keller

Beginner2021-11-19Added 14 answers

Step 1: Expand.

\int 4 x^{6} - 28 x^{3} + 49 d x

Step 2: Use Power Rule:

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

\frac{4 x^{7}}{7} - 7 x^{4} + 49 x

Step 3: Add constant.

\frac{4 x^{7}}{7} - 7 x^{4} + 49 x + C

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