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

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

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

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

Solution steps

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

Apply Integration By Parts

: 5^{x} sin (x) - \int 5^{x} ln (5) sin (x) d x

= 5^{x} sin (x) - \int 5^{x} ln (5) sin (x) d x

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

\int tan (- x) d x

\int \frac{16 x ^{2} - 1}{x ^{3}} d x

\int 2 e^{x} + x^{2} d x

\int \frac{1}{( 2 + sin ( x ) )} d x

\int \frac{x ^{2}}{( x - 1 ) ^{10}} d x

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