Math Problem Statement

∫ (x^2)/(sqrt(x^2 +5)) dx

Solution

To solve the integral

x2x2+5dx\int \frac{x^2}{\sqrt{x^2 + 5}} \, dx

we can use a substitution method. Here's the step-by-step process:

Step 1: Substitution

Let’s set: u=x2+5u = x^2 + 5 Now, differentiate both sides with respect to xx: du=2xdxdu = 2x \, dx

Thus: du2=xdx\frac{du}{2} = x \, dx

Now, rewrite the original integral in terms of uu:

x2x2+5dx=(u5)udu2x\int \frac{x^2}{\sqrt{x^2 + 5}} \, dx = \int \frac{(u - 5)}{\sqrt{u}} \cdot \frac{du}{2x}

This approach isn't simplifying easily. Let's try to apply integration by parts instead or look into trigonometric substitution

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Integration
Substitution Method
Trigonometric Substitution

Formulas

Substitution: u = x^2 + 5
du = 2x dx
Integration by parts: ∫u dv = uv - ∫v du

Theorems

Substitution Rule
Integration by Parts

Suitable Grade Level

Undergraduate Calculus