integral of 5e^x-cos(x)+3x^2-1/x

1) Let's compute the integral:

\int (5 e^{x} - \cos (x) + 3 x^{2} - \frac{1}{x}) d x

For the math expression:

- \cos (x)

, since (sin(a))' = a'*cos(a), we try differentiating

\sin (x)

2) Let's perform the differentiation:

\frac{d}{d x} (\sin (x))

ⓘ The derivative of sin(x) is cos(x):

\frac{d}{d x} (\sin (x))

\cos (x)

\cos (x)

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

ⅰ

\int (5 e^{x} - \cos (x) + 3 x^{2} - \frac{1}{x}) d x

- \sin (x) + \int (5 e^{x} + 3 x^{2} - \frac{1}{x}) d x

- \sin (x) + \int (5 e^{x} + 3 x^{2} - \frac{1}{x}) d x

For the math expression:

5 e^{x}

, since (exp(a))' = a'*exp(a), we try differentiating

\exp (x)

3) Let's perform the differentiation:

\frac{d}{d x} (\exp (x))

ⓘ The derivative of exp(x) is exp(x):

\frac{d}{d x} (\exp (x))

\exp (x)

\exp (x)

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

ⅰ

\int (5 e^{x} + 3 x^{2} - \frac{1}{x}) d x

5 e^{x} + \int (3 x^{2} - \frac{1}{x}) d x

- \sin (x) + (5 e^{x} + \int (3 x^{2} - \frac{1}{x}) d x)

ⅰ

\int (3 x^{2} - \frac{1}{x}) d x

\int 3 x^{2} d x - \int \frac{1}{x} d x

- \sin (x) + 5 e^{x} + (\int 3 x^{2} d x - \int \frac{1}{x} d x)

ⓘ A constant factor stays in front of the integral. An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

\int 3 x^{2} d x

3 \cdot \frac{x^{3}}{3}

■The derivative of

- \ln | x |

- \frac{1}{x}

, so

- \ln | x |

is an antiderivative of

- \frac{1}{x}

- \int \frac{1}{x} d x

- \ln | x |

- \sin (x)

+ 5 e^{x}

+ 3 \cdot \frac{x^{3}}{3}

- \ln | x |

ⓘ Multiplying a fraction by a number multiplies its numerator:

3 \cdot \frac{x^{3}}{3}

\frac{3 x^{3}}{3}

■A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{3 x^{3}}{3}

x^{3}

- \sin (x)

+ 5 e^{x}

+ x^{3}

- \ln | x |

▶Indefinite integrals are defined up to an additive constant C, so this yields:

5 e^{x}

+ x^{3}

- \ln | x |

- \sin (x)

+ C