What is the integral of (1+5x)cos(4x) ?

The integral of (1+5x)cos(4x) is 1/4 sin(4x)+5/16 (4xsin(4x)+cos(4x))+C

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integral of (1+5x)cos(4x)

\int (1 + 5 x) cos (4 x) d x

\frac{1}{4} sin (4 x) + \frac{5}{16} (4 x sin (4 x) + cos (4 x)) + C

Solution steps

\int (1 + 5 x) cos (4 x) d x

Expand

(1 + 5 x) cos (4 x) : cos (4 x) + 5 x cos (4 x)

= \int cos (4 x) + 5 x cos (4 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 cos (4 x) d x + \int 5 x cos (4 x) d x

\int cos (4 x) d x = \frac{1}{4} sin (4 x)

\int 5 x cos (4 x) d x = \frac{5}{16} (4 x sin (4 x) + cos (4 x))

= \frac{1}{4} sin (4 x) + \frac{5}{16} (4 x sin (4 x) + cos (4 x))

Add a constant to the solution

= \frac{1}{4} sin (4 x) + \frac{5}{16} (4 x sin (4 x) + cos (4 x)) + C

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What is the integral of (1+5x)cos(4x) ?
The integral of (1+5x)cos(4x) is 1/4 sin(4x)+5/16 (4xsin(4x)+cos(4x))+C