What is the integral of 1/(3x)+cos(2x)-5x^2 ?

The integral of 1/(3x)+cos(2x)-5x^2 is 1/3 ln|x|+1/2 sin(2x)-(5x^3)/3+C

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

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

\frac{1}{3} ln ∣ x ∣ + \frac{1}{2} sin (2 x) - \frac{5 x ^{3}}{3} + C

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{3 x} d x = \frac{1}{3} ln ∣ x ∣

\int cos (2 x) d x = \frac{1}{2} sin (2 x)

\int 5 x^{2} d x = \frac{5 x ^{3}}{3}

= \frac{1}{3} ln ∣ x ∣ + \frac{1}{2} sin (2 x) - \frac{5 x ^{3}}{3}

Add a constant to the solution

= \frac{1}{3} ln ∣ x ∣ + \frac{1}{2} sin (2 x) - \frac{5 x ^{3}}{3} + C

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What is the integral of 1/(3x)+cos(2x)-5x^2 ?
The integral of 1/(3x)+cos(2x)-5x^2 is 1/3 ln|x|+1/2 sin(2x)-(5x^3)/3+C