What is the integral of 6cos(sqrt(7x)) ?

The integral of 6cos(sqrt(7x)) is 12/7 (sqrt(7)sqrt(x)sin(sqrt(7)sqrt(x))+cos(sqrt(7)sqrt(x)))+C

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integral of 6cos(sqrt(7x))

\int 6 cos (7 x) d x

\frac{12}{7} (7 x sin (7 x) + cos (7 x)) + C

Solution steps

\int 6 cos (7 x) d x

Take the constant out:

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

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

Apply u-substitution

: \int cos (u) \frac{1}{7} d u

= 6 \cdot \int cos (u) \frac{1}{7} d u

Take the constant out:

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

= 6 \cdot \frac{1}{7} \cdot \int cos (u) d u

Apply u-substitution

: \int cos (v) \cdot 2 v d v

= 6 \cdot \frac{1}{7} \cdot \int cos (v) \cdot 2 v d v

Take the constant out:

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

= 6 \cdot \frac{1}{7} \cdot 2 \cdot \int cos (v) v d v

Apply Integration By Parts

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

= 6 \cdot \frac{1}{7} \cdot 2 (v sin (v) - \int sin (v) d v)

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

= 6 \cdot \frac{1}{7} \cdot 2 (v sin (v) - (- cos (v)))

Substitute back

= 6 \cdot \frac{1}{7} \cdot 2 (7 x sin (7 x) - (- cos (7 x)))

Simplify

6 \cdot \frac{1}{7} \cdot 2 (7 x sin (7 x) - (- cos (7 x))) : \frac{12}{7} (7 x sin (7 x) + cos (7 x))

= \frac{12}{7} (7 x sin (7 x) + cos (7 x))

Add a constant to the solution

= \frac{12}{7} (7 x sin (7 x) + cos (7 x)) + C

e^{s i n (x^{2})}

x \to - \infty lim (\frac{2 x ^{2} + 3}{2 x + 3})

x \to - 1 lim (3)

\int (24 x + 24) e^{4 x^{2} + 8 x + b} d x

\frac{d}{d x} ((x^{3} + 4) e^{3 x})

What is the integral of 6cos(sqrt(7x)) ?
The integral of 6cos(sqrt(7x)) is 12/7 (sqrt(7)sqrt(x)sin(sqrt(7)sqrt(x))+cos(sqrt(7)sqrt(x)))+C