QuestionJuly 14, 2025

Fill in the Blank 1 point int (4x)/(sqrt (1+x^2))dx=type your answer...

Fill in the Blank 1 point int (4x)/(sqrt (1+x^2))dx=type your answer...
Fill in the Blank 1 point int (4x)/(sqrt (1+x^2))dx=type your answer...

Solution
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Answer

4\sqrt{1 + x^2} + C Explanation 1. Use substitution Let u = 1 + x^2, then du = 2x \, dx. Therefore, 2x \, dx = du and dx = \frac{du}{2x}. 2. Simplify the integral Substitute into the integral: \int \frac{4x}{\sqrt{u}} \cdot \frac{du}{2x} = \int \frac{2}{\sqrt{u}} \, du. 3. Integrate The integral becomes 2 \int u^{-\frac{1}{2}} \, du = 2 \cdot 2u^{\frac{1}{2}} + C = 4\sqrt{u} + C. 4. Back-substitute Replace u with 1 + x^2: 4\sqrt{1 + x^2} + C.

Explanation

1. Use substitution<br /> Let $u = 1 + x^2$, then $du = 2x \, dx$. Therefore, $2x \, dx = du$ and $dx = \frac{du}{2x}$.<br />2. Simplify the integral<br /> Substitute into the integral: $\int \frac{4x}{\sqrt{u}} \cdot \frac{du}{2x} = \int \frac{2}{\sqrt{u}} \, du$.<br />3. Integrate<br /> The integral becomes $2 \int u^{-\frac{1}{2}} \, du = 2 \cdot 2u^{\frac{1}{2}} + C = 4\sqrt{u} + C$.<br />4. Back-substitute<br /> Replace $u$ with $1 + x^2$: $4\sqrt{1 + x^2} + C$.
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