Resolved: "Evaluate the indefinite integral. ∫3x+25x6dx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Cynthia Bell

Answered question

2021-12-27

Evaluate the indefinite integral.

\int \frac{3}{x} + \frac{2}{5 x^{6}} d x

Answer & Explanation

censoratojk

Beginner2021-12-28Added 46 answers

Step 1
This question is based on indefinite integral.
General form,
Integral of

x^{n}

\frac{x^{(x + 1)}}{n + 1}

As for x = 1 formula violates because result comes is not defined.
Therefore, integral of 1/x is

\ln (x)

Step 2
Given

\int (\frac{3}{x} + \frac{2}{5 x^{6}}) d x

= \int \frac{3}{x} d x + \int \frac{2}{5 x^{6}} d x ∵ \int x^{n} d x = \frac{x^{n + 1}}{n + 1}

and

\int \frac{1}{x} d x = \ln (x)

= 3 \ln (x) + \frac{2}{5} \frac{x^{(- 6 + 1)}}{(- 6 + 1)} + C

= 3 \ln (x) + \frac{2}{5} \frac{x^{- 5}}{- 5} + C

= 3 \ln (x) - \frac{2}{25} x^{- 5} + C

Hence, Answer is

3 \ln (x) - \frac{2}{25} x^{- 5} + C

Foreckije

Beginner2021-12-29Added 32 answers

\int (\frac{3}{x} + \frac{2}{5 \cdot x^{6}}) d x

Lets

karton

Expert2022-01-04Added 613 answers

Given:
$\int \frac{3}{x} + \frac{2}{5 x^{6}} d x$
Use properties of integrals
$\int \frac{3}{x} d x + \int \frac{2}{5 x^{6}} d x$
Evaluate the integrals
$3 \ln (| x |) - \frac{2}{25 x^{5}}$
Add C
Answer:
$3 \ln (| x |) - \frac{2}{25 x^{5}} + C$

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