5 Multiple Choice 1 point \( \int(1 / 3) x^{4} d x \) \( \int x^{2} \sin (10 x) d x \) \( \int x \sqrt{ }\left(5+x^{2}\right) d x \) \( \int x^{-1} d x \)

Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int x^{-1} dx\) - step1: Rewrite the expression: \(\int \frac{1}{x} dx\) - step2: Evaluate the integral: \(\ln{\left(\left|x\right|\right)}\) - step3: Add the constant of integral C: \(\ln{\left(\left|x\right|\right)} + C, C \in \mathbb{R}\) Calculate the integral \( \int x \sqrt{5+x^{2}} d x \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int x\sqrt{5+x^{2}} dx\) - step1: Rewrite the expression: \(\int x\left(5+x^{2}\right)^{\frac{1}{2}} dx\) - step2: Use the substitution \(dx=\frac{1}{2x} dt\) to transform the integral\(:\) \(\int x\left(5+x^{2}\right)^{\frac{1}{2}}\times \frac{1}{2x} dt\) - step3: Simplify: \(\int \frac{\left(5+x^{2}\right)^{\frac{1}{2}}}{2} dt\) - step4: Use the substitution \(t=x^{2}\) to transform the integral\(:\) \(\int \frac{\left(5+t\right)^{\frac{1}{2}}}{2} dt\) - step5: Rewrite the expression: \(\int \frac{1}{2}\left(5+t\right)^{\frac{1}{2}} dt\) - step6: Use properties of integrals: \(\frac{1}{2}\times \int \left(5+t\right)^{\frac{1}{2}} dt\) - step7: Use the substitution \(dt=1 dv\) to transform the integral\(:\) \(\frac{1}{2}\times \int \left(5+t\right)^{\frac{1}{2}}\times 1 dv\) - step8: Simplify: \(\frac{1}{2}\times \int \left(5+t\right)^{\frac{1}{2}} dv\) - step9: Use the substitution \(v=5+t\) to transform the integral\(:\) \(\frac{1}{2}\times \int v^{\frac{1}{2}} dv\) - step10: Evaluate the integral: \(\frac{1}{2}\times \frac{v^{\frac{1}{2}+1}}{\frac{1}{2}+1}\) - step11: Simplify: \(\frac{1}{2}\times \frac{v^{\frac{3}{2}}}{\frac{3}{2}}\) - step12: Multiply the terms: \(\frac{v^{\frac{3}{2}}}{2\times \frac{3}{2}}\) - step13: Multiply the terms: \(\frac{v^{\frac{3}{2}}}{3}\) - step14: Substitute back: \(\frac{\left(5+t\right)^{\frac{3}{2}}}{3}\) - step15: Substitute back: \(\frac{\left(5+x^{2}\right)^{\frac{3}{2}}}{3}\) - step16: Simplify: \(\frac{\sqrt{\left(5+x^{2}\right)^{3}}}{3}\) - step17: Add the constant of integral C: \(\frac{\sqrt{\left(5+x^{2}\right)^{3}}}{3} + C, C \in \mathbb{R}\) Calculate the integral \( \int \frac{1}{3} x^{4} d x \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \frac{1}{3}x^{4} dx\) - step1: Use properties of integrals: \(\frac{1}{3}\times \int x^{4} dx\) - step2: Evaluate the integral: \(\frac{1}{3}\times \frac{x^{4+1}}{4+1}\) - step3: Simplify: \(\frac{1}{3}\times \frac{x^{5}}{5}\) - step4: Multiply the terms: \(\frac{x^{5}}{3\times 5}\) - step5: Multiply the terms: \(\frac{x^{5}}{15}\) - step6: Add the constant of integral C: \(\frac{x^{5}}{15} + C, C \in \mathbb{R}\) Calculate the integral \( \int x^{2} \sin(10 x) d x \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int x^{2}\sin\left(10x\right) dx\) - step1: Prepare for integration by parts: \(\begin{align}&u=x^{2}\\&dv=\sin\left(10x\right)dx\end{align}\) - step2: Calculate the derivative: \(\begin{align}&du=2xdx\\&dv=\sin\left(10x\right)dx\end{align}\) - step3: Evaluate the integral: \(\begin{align}&du=2xdx\\&v=-\frac{1}{10}\cos\left(10x\right)\end{align}\) - step4: Substitute the values into formula: \(x^{2}\left(-\frac{1}{10}\cos\left(10x\right)\right)-\int 2x\left(-\frac{1}{10}\cos\left(10x\right)\right) dx\) - step5: Calculate: \(-\frac{1}{10}x^{2}\cos\left(10x\right)-\int -\frac{1}{5}x\cos\left(10x\right) dx\) - step6: Evaluate the integral: \(-\frac{1}{10}x^{2}\cos\left(10x\right)+\frac{1}{50}x\sin\left(10x\right)+\frac{1}{500}\cos\left(10x\right)\) - step7: Add the constant of integral C: \(-\frac{1}{10}x^{2}\cos\left(10x\right)+\frac{1}{50}x\sin\left(10x\right)+\frac{1}{500}\cos\left(10x\right) + C, C \in \mathbb{R}\) Here are the results for the integrals you provided: 1. **Integral of \( \frac{1}{3} x^{4} \)**: \[ \int \frac{1}{3} x^{4} d x = \frac{x^{5}}{15} + C, \quad C \in \mathbb{R} \] 2. **Integral of \( x^{2} \sin(10 x) \)**: \[ \int x^{2} \sin(10 x) d x = -\frac{1}{10}x^{2}\cos(10x) + \frac{1}{50}x\sin(10x) + \frac{1}{500}\cos(10x) + C, \quad C \in \mathbb{R} \] 3. **Integral of \( x \sqrt{5+x^{2}} \)**: \[ \int x \sqrt{5+x^{2}} d x = \frac{\sqrt{(5+x^{2})^{3}}}{3} + C, \quad C \in \mathbb{R} \] 4. **Integral of \( x^{-1} \)**: \[ \int x^{-1} d x = \ln{(|x|)} + C, \quad C \in \mathbb{R} \] These results represent the indefinite integrals of the given functions, with \( C \) being the constant of integration.