Step by step solution to "Given ∫(x5−3x4+6x3+3)dx, evaluate the indefinite integral."

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Concepcion Hale

Answered question

2022-01-07

Given

\int (x^{5} - 3 x^{4} + 6 x^{3} + 3) d x

, evaluate the indefinite integral.

Answer & Explanation

Beverly Smith

Beginner2022-01-08Added 42 answers

Step 1
The given integral is

\int (x^{5} - 3 x^{4} + 6 x^{3} + 3) d x

.
obtain the integral value as follows.

\int (x^{5} - 3 x^{4} + 6 x^{3} + 3) d x = \int x^{5} d x - \int 3 x^{4} d x + \int 6 x^{3} d x + \int 3 d x

= \frac{x^{6}}{6} - 3 (\frac{x^{5}}{5}) + 6 (\frac{x^{4}}{4}) + 3 x + C

= \frac{x^{6}}{6} - \frac{3 x^{5}}{5} + \frac{3 x^{4}}{2} + 3 x + C

Thus, the value of the integral is

\frac{x^{6}}{6} - \frac{3 x^{5}}{5} + \frac{3 x^{4}}{2} + 3 x

Step 2
Answer:
The value of the given integral is

(\frac{x^{6}}{6}) - 3 (\frac{x^{5}}{5}) + 3 (\frac{x^{4}}{2}) + 3 x

ambarakaq8

Beginner2022-01-09Added 31 answers

Given integral.

\int x^{5} - 3 x^{4} + 6 x^{3} + 3 d x

\int x^{5} d x - \int 3 x^{4} d x + \int 6 x^{3} d x + \int 3 d x

Evaluate

\frac{x^{6}}{6} - \frac{3 x^{5}}{5} + \frac{3 x^{4}}{2} + 3 x

Add C
Answer:

\frac{x^{6}}{6} - \frac{3 x^{5}}{5} + \frac{3 x^{4}}{2} + 3 x + C

karton

Expert2022-01-11Added 613 answers

$I = \int (x^{5} - 3 x^{4} + 6 x^{3} + 3) d x we know that \int x^{n} d x = \frac{x^{n + 1}}{n + 1} I = \frac{x^{5 + 1}}{5 + 1} - \frac{3 x^{4 + 1}}{4 + 1} + \frac{6 x^{3 + 1}}{3 + 1} + 3 x I = \frac{x^{6}}{6} - \frac{3 x^{5}}{5} + \frac{6 x^{4}}{4} + 3 x I = \frac{x^{6}}{6} - \frac{3}{5} x^{5} + \frac{3}{2} x^{4} + 3 x$

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