∫sqrt[3]x (x^3+4x^2+7) dx is equal to-

The correct answer is Option (2) → \( 3\left[\displaystyle\frac{1}{13}x^{\frac{13}{3}}+\frac{2}{5}x^{\frac{10}{3}}+\frac{7}{4}x^{\frac{4}{3}}\right]+c \), where C is an arbitrary constant.

Since \( \sqrt[3]{x}=x^{1/3} \), the integrand multiplies out into powers of x, and each is integrated by \( \int x^n\,dx=\frac{x^{n+1}}{n+1} \).

    \( \displaystyle x^{1/3}(x^3+4x^2+7)=x^{10/3}+4x^{7/3}+7x^{1/3} \)

    \( \displaystyle \int\left(x^{10/3}+4x^{7/3}+7x^{1/3}\right)dx=\frac{x^{13/3}}{13/3}+4\cdot\frac{x^{10/3}}{10/3}+7\cdot\frac{x^{4/3}}{4/3}+c \)

    \( \displaystyle =\frac{3}{13}x^{13/3}+\frac{12}{10}x^{10/3}+\frac{21}{4}x^{4/3}+c \)

Taking 3 out as a common factor:

    \( \displaystyle =3\left[\frac{1}{13}x^{13/3}+\frac{2}{5}x^{10/3}+\frac{7}{4}x^{4/3}\right]+c \)

Why the other options are incorrect

• Option (1) \( 3\left[13x^{\frac{3}{13}}+\displaystyle\frac{5}{2}x^{\frac{3}{10}}+\frac{4}{7}x^{\frac{3}{4}}\right]+c \), where C is an arbitrary constant. — Incorrect. The powers are turned upside down (\( \frac{3}{13} \) instead of \( \frac{13}{3} \)), and so are the coefficients.

• Option (3) \( \displaystyle\frac{1}{3}\left[\frac{1}{13}x^{13/3}+\frac{5}{2}x^{10/3}+\frac{7}{4}x^{4/3}\right]+c \), where C is an arbitrary constant. — Incorrect. The factor outside should be 3, not \( \frac{1}{3} \), and the middle coefficient should be \( \frac{2}{5} \), not \( \frac{5}{2} \).

• Option (4) \( 3\left[13x^{\frac{3}{13}}+\displaystyle\frac{2}{5}x^{\frac{10}{3}}+\frac{4}{7}x^{\frac{4}{3}}\right]+c \), where C is an arbitrary constant. — Incorrect. The first and last terms are wrong: the powers should be \( \frac{13}{3} \) and \( \frac{4}{3} \) with coefficients \( \frac{1}{13} \) and \( \frac{7}{4} \) inside the bracket.