Get the answer to "Evaluate the indefinite integral. x3+3x−4x2dx"

Maaghu

Answered question

2021-11-20

Evaluate the indefinite integral.

\frac{x^{3} + 3 x - 4}{x^{2}} d x

Upout1940

Beginner2021-11-21Added 9 answers

Step 1
Given

\frac{x^{3} + 3 x - 4}{x^{2}} d x

By dividing each term of the numerator by the denominator and separate the integral

= \int \frac{x^{3}}{x^{2}} d x + \int \frac{3 x}{x^{2}} d x - \int \frac{4}{x^{2}} d x

= \int x d x + \int \frac{3}{x} d x - \int \frac{4}{x^{2}} d x

Step 2
Now we take integration separately.

= \int x d x + 3 \int \frac{1}{x} d x - 4 \int x^{- 2} d x

= \frac{x^{2}}{2} + 4 \ln x + \frac{4}{x} + C

This is the required integral.

pendukke

Beginner2021-11-22Added 9 answers

Step 1: Simplify

\frac{x^{3} + 3 x - 4}{x^{2}} \to x + \frac{3 x - 4}{x^{2}}

\int x + \frac{3 x - 4}{x^{2}} 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 x d x + \int \frac{3 x - 4}{x^{2}} d x

Step 3: Use Power Rule:

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

\frac{x^{2}}{2} + \int \frac{3 x - 4}{x^{2}} d x

Step 4: Split fraction.

\frac{x^{2}}{2} + \int \frac{3 x}{x^{2}} + \frac{- 4}{x^{2}} d x

Step 5: Use Sum Rule:

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

\frac{x^{2}}{2} + \int \frac{3}{x} d x + \int - \frac{4}{x^{2}} d x

Step 6: Use Constant Factor Rule:

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

\frac{x^{2}}{2} + 3 \int \frac{1}{x} d x + \int - \frac{4}{x^{2}} d x

Step 7: The derivative of

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

\frac{x^{2}}{2} + 3 \ln x + \int - \frac{4}{x^{2}} d x

Step 8: Use Constant Factor Rule:

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

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