We have an answer to "Integral (cos⁡(x)+14x)dx to find the general indefinite integral."

Steven Smith

Answered question

2021-11-19

Integral

(\cos (x) + \frac{1}{4} x) d x

to find the general indefinite integral.

Anthony Caraballo

Beginner2021-11-20Added 15 answers

Step 1
Given integral,

I = \int (\cos x + \frac{1}{4 x}) d x

Used formulas,

\int (f (x) + g (x)) d x = \int f (x) d x + \int g (x) d x

\int \cos x d x = \sin x

\int k f (x) = k \int f (x)

{k is constant}

\int \frac{1}{x} d x = \ln | x |

Step 2

I = \int (\cos x + \frac{1}{4 x}) d x

I = \int \cos x d x + \int \frac{1}{4 x} d x

I = \sin x + \frac{1}{4} \int \frac{1}{x} d x

I = \sin x + \frac{1}{4} \ln | x | + c

Step 3
Therefore,

\int (\cos x + \frac{1}{4 x}) d x = \sin x + \frac{1}{4} \ln | x | + c

where c is constant of integration.

Lorraine David

Beginner2021-11-21Added 13 answers

Step 1: Expand.

\int \cos x + \frac{1}{4 x} d x

Step 2: Use Sum Rule:

\int f (x) + g (x) d x = \int f (x) d x + \int g (x) d x

\int \cos x d x + \int \frac{1}{4 x} d x

Step 3: Use Trigonometric Integration: the integral of

\cos x i s \sin x

\sin x + \int \frac{1}{4 x} d x

Step 4: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

\sin x + \frac{1}{4} \int \frac{1}{x} d x

Step 5: The derivative of

\ln x i s \frac{1}{x}

\sin x + \frac{\ln x}{4}

Step 6: Add constant.

\sin x + \frac{\ln x}{4} + C

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