What is the integral of x^3+2/(5-6x)+3cos(5-7x) ?

The integral of x^3+2/(5-6x)+3cos(5-7x) is (x^4)/4-1/3 ln|5-6x|-3/7 sin(5-7x)+C

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integral of x^3+2/(5-6x)+3cos(5-7x)

\int x^{3} + \frac{2}{5 - 6 x} + 3 cos (5 - 7 x) d x

\frac{x ^{4}}{4} - \frac{1}{3} ln ∣ 5 - 6 x ∣ - \frac{3}{7} sin (5 - 7 x) + C

Solution steps

\int x^{3} + \frac{2}{5 - 6 x} + 3 cos (5 - 7 x) 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 x^{3} d x + \int \frac{2}{5 - 6 x} d x + \int 3 cos (5 - 7 x) d x

\int x^{3} d x = \frac{x ^{4}}{4}

\int \frac{2}{5 - 6 x} d x = - \frac{1}{3} ln ∣ 5 - 6 x ∣

\int 3 cos (5 - 7 x) d x = - \frac{3}{7} sin (5 - 7 x)

= \frac{x ^{4}}{4} - \frac{1}{3} ln ∣ 5 - 6 x ∣ - \frac{3}{7} sin (5 - 7 x)

Add a constant to the solution

= \frac{x ^{4}}{4} - \frac{1}{3} ln ∣ 5 - 6 x ∣ - \frac{3}{7} sin (5 - 7 x) + C

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What is the integral of x^3+2/(5-6x)+3cos(5-7x) ?
The integral of x^3+2/(5-6x)+3cos(5-7x) is (x^4)/4-1/3 ln|5-6x|-3/7 sin(5-7x)+C