QuestionFind the integral: $\int \frac{5-x-3 x^{2}}{4 \sqrt{x}} \, dx$

Studdy Solution

STEP 1

Assumptions1. We are given the integral $\int \frac{5-x-3 x^{}}{4 \sqrt{x}}$ . We need to find the antiderivative of this function3. We will use the power rule for integration, which states that $\int x^n dx = \frac{1}{n+1}x^{n+1} + C$ where C is the constant of integration

STEP 2

First, we can simplify the integral by dividing each term in the numerator by $4\sqrt{x}$ . This will give us three separate integrals.
$\int \frac{5-x- x^{2}}{4 \sqrt{x}} dx = \int \frac{5}{4\sqrt{x}} dx - \int \frac{x}{4\sqrt{x}} dx - \int \frac{x^2}{4\sqrt{x}} dx$

STEP 3

Next, we simplify each integral by rewriting $\sqrt{x}$ as $x^{1/2}$ .
$\int \frac{5}{x^{1/2}} dx - \int \frac{x}{x^{1/2}} dx - \int \frac{3x^2}{x^{1/2}} dx$

STEP 4

We can further simplify each integral by dividing the terms.
$\int \frac{}{4}x^{-1/2} dx - \int \frac{1}{4}x^{1/2} dx - \int \frac{3}{4}x^{3/2} dx$

STEP 5

Now, we can integrate each term separately using the power rule for integration.
$\frac{5}{4}\int x^{-1/2} dx - \frac{1}{4}\int x^{1/2} dx - \frac{3}{4}\int x^{3/2} dx$

STEP 6

Applying the power rule for integration, we get $\frac{5}{4} \cdot2x^{1/2} - \frac{1}{4} \cdot \frac{2}{3}x^{3/2} - \frac{3}{4} \cdot \frac{2}{5}x^{5/2} + C$

STEP 7

implify the expression to get the final answer.
$\frac{5}{2}x^{1/2} - \frac{1}{6}x^{3/2} - \frac{3}{10}x^{5/2} + C$ So, the antiderivative of $\int \frac{5-x-3 x^{2}}{4 \sqrt{x}} dx$ is $\frac{5}{2}x^{1/2} - \frac{1}{6}x^{3/2} - \frac{3}{10}x^{5/2} + C$ .

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