Solutions

Find the following indefinite integrals:

• $\int (x^2 - 3x + 3 - \frac{1}{x})dx$.
  $$\int (x^2 - 3x + 3 - \frac{1}{x})dx = \frac{x^3}{3} - \frac{3x^2}{2} + 3x - \ln(\vert x\vert) + C.$$

• $\int (\cos(t) - \sqrt{t}) dt$.
  $$\int (\cos(t) - \sqrt{t}) dt = \sin(t) - \frac{2}{3}t^{\frac{3}{2}} + C.$$

• $\int (\sec(t)\tan(t) - \sec^2(t))dt$.
  $$\int (\sec(t)\tan(t) - \sec^2(t)) dt = \sec(t) - \tan(t) + C.$$

• $\int (2^t + 3^t + 4^t)dt$.
  $$\int (2^t + 3^t + 4^t)dt = \frac{2^t}{\ln(2)} + \frac{3^t}{\ln(3)} + \frac{4^t}{\ln(4)} + C.$$

• $\int (t - \frac{1}{t})^2 dt$.
  $$\int (t - \frac{1}{t})^2 dt = \int (t^2 - 2 + \frac{1}{t^2}) dt = \frac{1}{3}t^3 - 2t -\frac{1}{t} + C.$$

• $\int (x +\sqrt{x})^2 dx$.
  $$\int (x +\sqrt{x})^2 dx = \int (x^2 + 2x^{\frac{3}{2}} + x) dx = \frac{1}{3}x^3 + \frac{4}{5}x^{\frac{5}{2}} + \frac{1}{2}x^2 + C.$$

• $\int (4e^x - \sin(x))dx$,
  $$\int (4e^x - \sin(x))dx = 4e^x + \cos(x) + C.$$

• $\int \sin(t - 2) dt$.
  $$\int \sin(t - 2) dt = \int (\sin(t)\cos(2) - \cos(t) \sin(2))dt$$
  $$= -\cos(t)\cos(2) - \sin(t)\sin(2) + C = - \cos(t - 2) + C.$$

• $\int (x^{\frac{1}{3}} + x^{-\frac{2}{3}}) dx$.
  $$\int (x^{\frac{1}{3}} + x^{-\frac{2}{3}}) dx = \frac{3x^{\frac{4}{3}}}{4} + 3x^{\frac{1}{3}} + C.$$

• $\int \frac{x^3 + 1}{x(x + 1)} dx$.
  $$\int \frac{x^3 + 1}{x(x + 1)} dx = \int \frac{(x + 1)(x^2 - x + 1)}{x(x + 1)}dx = \int \frac{x^2 - x + 1}{x}dx$$
  $$= \int (x - 1 + \frac{1}{x})dx =\frac{1}{2}x^2 - x + \ln(\vert x\vert) + C.$$