What is the integral of 7xcos(6x) ?

The integral of 7xcos(6x) is 7/36 (6xsin(6x)+cos(6x))+C

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integral of 7xcos(6x)

\int 7 x cos (6 x) d x

\frac{7}{36} (6 x sin (6 x) + cos (6 x)) + C

Solution steps

\int 7 x cos (6 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 7 \cdot \int x cos (6 x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{36} d u

= 7 \cdot \int \frac{u cos ( u )}{36} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 7 \cdot \frac{1}{36} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= 7 \cdot \frac{1}{36} (u sin (u) - \int sin (u) d u)

\int sin (u) d u = - cos (u)

= 7 \cdot \frac{1}{36} (u sin (u) - (- cos (u)))

Substitute back

u = 6 x

= 7 \cdot \frac{1}{36} (6 x sin (6 x) - (- cos (6 x)))

Simplify

7 \cdot \frac{1}{36} (6 x sin (6 x) - (- cos (6 x))) : \frac{7}{36} (6 x sin (6 x) + cos (6 x))

= \frac{7}{36} (6 x sin (6 x) + cos (6 x))

Add a constant to the solution

= \frac{7}{36} (6 x sin (6 x) + cos (6 x)) + C

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What is the integral of 7xcos(6x) ?
The integral of 7xcos(6x) is 7/36 (6xsin(6x)+cos(6x))+C