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kdgg0909gn

Answered question

2021-11-18

Find the indefinite integral and check the result by differentiation.

\int (x^{5} + 1) d x

Cherry McCormick

Beginner2021-11-19Added 23 answers

Step 1
Use the power rule of integration

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

and integrate the function

x^{5} + 1

.
Step 2

\int (x^{5} + 1) d x = \frac{x^{6}}{6} + x + c

Step 3
Now differentiate the function

\frac{x^{6}}{6} + x + c

with respect to x.

\frac{d}{d x} (\frac{x^{6}}{6} + x + c) = \frac{1}{6} (6 x^{5}) + 1 + 0

= x^{5} + 1

As the derivative of the function

\frac{x^{6}}{6} + x + c i s x^{5} + 1

, the integral is verified.
Step 4
Conclusion:

\int (x^{5} + 1) d x = \frac{x^{6}}{6} + x + c

As the derivative of the function

\frac{x^{6}}{6} + x + c i s x^{5} + 1

, the integral is verified.

Poul1963

Beginner2021-11-20Added 15 answers

Step 1: Expand.

\int x^{5} + 1 d x

Step 2: Use Power Rule:

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

\frac{x^{6}}{6} + x

Step 3: Add constant.

\frac{x^{6}}{6} + x + C

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