Evaluate (a) \( \int\left(x^{2}+10\right)^{50} 2 x d x \)

Calculate the integral \( \int\left(x^{2}+10\right)^{50} 2 x d x \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \left(x^{2}+10\right)^{50}\times 2x dx\) - step1: Multiply the terms: \(\int 2x\left(x^{2}+10\right)^{50} dx\) - step2: Use properties of integrals: \(2\times \int x\left(x^{2}+10\right)^{50} dx\) - step3: Use the substitution \(dx=\frac{1}{2x} dt\) to transform the integral\(:\) \(2\times \int x\left(x^{2}+10\right)^{50}\times \frac{1}{2x} dt\) - step4: Simplify: \(2\times \int \frac{1}{2}\left(x^{2}+10\right)^{50} dt\) - step5: Use the substitution \(t=x^{2}\) to transform the integral\(:\) \(2\times \int \frac{1}{2}\left(t+10\right)^{50} dt\) - step6: Use properties of integrals: \(\int \left(t+10\right)^{50} dt\) - step7: Use the substitution \(v=t+10\) to transform the integral\(:\) \(\int v^{50} dv\) - step8: Evaluate the integral: \(\frac{v^{51}}{51}\) - step9: Simplify: \(\frac{1}{51}v^{51}\) - step10: Substitute back: \(\frac{1}{51}\left(t+10\right)^{51}\) - step11: Substitute back: \(\frac{1}{51}\left(x^{2}+10\right)^{51}\) - step12: Add the constant of integral C: \(\frac{1}{51}\left(x^{2}+10\right)^{51} + C, C \in \mathbb{R}\) The result of the integral \( \int\left(x^{2}+10\right)^{50} 2 x d x \) is \( \frac{1}{51}(x^{2}+10)^{51} + C \), where \( C \) is an arbitrary constant.