Washington Whittaker
04/23/2024 · Senior High School
\( \int \frac { x d x } { \sqrt[ 4 ] { x ^ { 2 } + 1 } } d x \)
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Step-by-step Solution
Calculate the integral \( \int \frac { x d x } { \sqrt[ 4 ] { x ^ { 2 } + 1 } } \).
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
\(\int \frac{x}{\sqrt[4]{x^{2}+1}} dx\)
- step1: Evaluate the power:
\(\int \frac{x}{\left(x^{2}+1\right)^{\frac{1}{4}}} dx\)
- step2: Use the substitution \(dx=\frac{1}{2x} dt\) to transform the integral\(:\)
\(\int \frac{x}{\left(x^{2}+1\right)^{\frac{1}{4}}}\times \frac{1}{2x} dt\)
- step3: Simplify:
\(\int \frac{1}{2\left(x^{2}+1\right)^{\frac{1}{4}}} dt\)
- step4: Use the substitution \(t=x^{2}\) to transform the integral\(:\)
\(\int \frac{1}{2\left(t+1\right)^{\frac{1}{4}}} dt\)
- step5: Use properties of integrals:
\(\frac{1}{2}\times \int \frac{1}{\left(t+1\right)^{\frac{1}{4}}} dt\)
- step6: Use the substitution \(v=t+1\) to transform the integral\(:\)
\(\frac{1}{2}\times \int \frac{1}{v^{\frac{1}{4}}} dv\)
- step7: Evaluate the integral:
\(\frac{1}{2}\times \frac{4v^{\frac{3}{4}}}{3}\)
- step8: Calculate:
\(\frac{2v^{\frac{3}{4}}}{3}\)
- step9: Substitute back:
\(\frac{2\left(t+1\right)^{\frac{3}{4}}}{3}\)
- step10: Substitute back:
\(\frac{2\left(x^{2}+1\right)^{\frac{3}{4}}}{3}\)
- step11: Simplify:
\(\frac{2\sqrt[4]{\left(x^{2}+1\right)^{3}}}{3}\)
- step12: Add the constant of integral C:
\(\frac{2\sqrt[4]{\left(x^{2}+1\right)^{3}}}{3} + C, C \in \mathbb{R}\)
La integral de \( \int \frac { x d x } { \sqrt[ 4 ] { x ^ { 2 } + 1 } } \) es igual a \( \frac{2\sqrt[4]{(x^{2}+1)^{3}}}{3} + C \), donde \( C \) es una constante real.
Quick Answer
La integral es \( \frac{2\sqrt[4]{(x^{2}+1)^{3}}}{3} + C \).
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