What is the integral of-5^xcos(7x) ?

The integral of-5^xcos(7x) is 7/(-49-ln^2(5))*5^xsin(7x)+1/(-49-ln^2(5))ln(5)*5^xcos(7x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of-5^xcos(7x)

\int - 5^{x} cos (7 x) d x

\frac{7}{- 49 - ln ^{2} ( 5 )} \cdot 5^{x} sin (7 x) + \frac{1}{- 49 - ln ^{2} ( 5 )} ln (5) \cdot 5^{x} cos (7 x) + C

Solution steps

\int - 5^{x} cos (7 x) d x

Take the constant out:

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

= - \int 5^{x} cos (7 x) d x

Apply Integration By Parts

: 5^{x} \cdot \frac{1}{7} sin (7 x) - \int 5^{x} ln (5) \frac{1}{7} sin (7 x) d x

= - (5^{x} \cdot \frac{1}{7} sin (7 x) - \int 5^{x} ln (5) \frac{1}{7} sin (7 x) d x)

Take the constant out:

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

= - (5^{x} \cdot \frac{1}{7} sin (7 x) - ln (5) \frac{1}{7} \cdot \int 5^{x} sin (7 x) d x)

Apply Integration By Parts

: - \frac{1}{7} \cdot 5^{x} cos (7 x) - \int - \frac{1}{7} ln (5) \cdot 5^{x} cos (7 x) d x

= - (5^{x} \cdot \frac{1}{7} sin (7 x) - ln (5) \frac{1}{7} (- \frac{1}{7} \cdot 5^{x} cos (7 x) - \int - \frac{1}{7} ln (5) \cdot 5^{x} cos (7 x) d x))

Take the constant out:

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

= - (5^{x} \cdot \frac{1}{7} sin (7 x) - ln (5) \frac{1}{7} (- \frac{1}{7} \cdot 5^{x} cos (7 x) - (- \frac{1}{7} ln (5) \cdot \int 5^{x} cos (7 x) d x)))

Therefore

- \int 5^{x} cos (7 x) d x = - (5^{x} \cdot \frac{1}{7} sin (7 x) - ln (5) \frac{1}{7} (- \frac{1}{7} \cdot 5^{x} cos (7 x) - (- \frac{1}{7} ln (5) \cdot \int 5^{x} cos (7 x) d x)))

Isolate

\int 5^{x} cos (7 x) d x

= - (- \frac{7 \cdot 5 ^{x} sin ( 7 x )}{- 49 - ln ^{2} ( 5 )} - \frac{ln ( 5 ) \cdot 5 ^{x} cos ( 7 x )}{- 49 - ln ^{2} ( 5 )})

Simplify

- (- \frac{7 \cdot 5 ^{x} sin ( 7 x )}{- 49 - ln ^{2} ( 5 )} - \frac{ln ( 5 ) \cdot 5 ^{x} cos ( 7 x )}{- 49 - ln ^{2} ( 5 )}) : \frac{7}{- 49 - ln ^{2} ( 5 )} \cdot 5^{x} sin (7 x) + \frac{1}{- 49 - ln ^{2} ( 5 )} ln (5) \cdot 5^{x} cos (7 x)

= \frac{7}{- 49 - ln ^{2} ( 5 )} \cdot 5^{x} sin (7 x) + \frac{1}{- 49 - ln ^{2} ( 5 )} ln (5) \cdot 5^{x} cos (7 x)

Add a constant to the solution

= \frac{7}{- 49 - ln ^{2} ( 5 )} \cdot 5^{x} sin (7 x) + \frac{1}{- 49 - ln ^{2} ( 5 )} ln (5) \cdot 5^{x} cos (7 x) + C

\int \frac{- 2}{x ( ln ( x ) ) ^{2}} d x

\int \frac{3 ln ( x ^{2} )}{x} d x

\int \frac{1 - x}{1 + 4 x ^{2}} d x

\int \frac{x}{x + 10} d x

\int \frac{1}{sin ^{2} ( w ) 1 - cot ( w )} d w

What is the integral of-5^xcos(7x) ?
The integral of-5^xcos(7x) is 7/(-49-ln^2(5))*5^xsin(7x)+1/(-49-ln^2(5))ln(5)*5^xcos(7x)+C