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Find the integral $\int e^x\cos\left(x\right)dx$

Step-by-step Solution

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Solution

Formulas

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Final answer to the problem

$\frac{1}{2}e^x\cos\left(x\right)+\frac{1}{2}e^x\sin\left(x\right)+C_0$
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Step-by-step Solution

How should I solve this problem?

  • Integrate by substitution
  • Integrate by partial fractions
  • Integrate by parts
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
  • FOIL Method
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Can't find a method? Tell us so we can add it.
1

We can solve the integral $\int e^x\cos\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
2

First, identify or choose $u$ and calculate it's derivative, $du$

$\begin{matrix}\displaystyle{u=\cos\left(x\right)}\\ \displaystyle{du=-\sin\left(x\right)dx}\end{matrix}$
3

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=e^xdx}\\ \displaystyle{\int dv=\int e^xdx}\end{matrix}$
4

Solve the integral to find $v$

$v=\int e^xdx$
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5

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$e^x$
6

Now replace the values of $u$, $du$ and $v$ in the last formula

$e^x\cos\left(x\right)+\int e^x\sin\left(x\right)dx$
7

The integral $\int e^x\sin\left(x\right)dx$ results in: $e^x\sin\left(x\right)-\int e^x\cos\left(x\right)dx$

$e^x\sin\left(x\right)-\int e^x\cos\left(x\right)dx$
8

This integral by parts turned out to be a cyclic one (the integral that we are calculating appeared again in the right side of the equation). We can pass it to the left side of the equation with opposite sign

$\int e^x\cos\left(x\right)dx=e^x\cos\left(x\right)-\int e^x\cos\left(x\right)dx+e^x\sin\left(x\right)$
9

Moving the cyclic integral to the left side of the equation

$\int e^x\cos\left(x\right)dx+\int e^x\cos\left(x\right)dx=e^x\cos\left(x\right)+e^x\sin\left(x\right)$
10

Adding the integrals

$2\int e^x\cos\left(x\right)dx=e^x\cos\left(x\right)+e^x\sin\left(x\right)$
11

Move the constant term $2$ dividing to the other side of the equation

$\int e^x\cos\left(x\right)dx=\frac{1}{2}\left(e^x\cos\left(x\right)+e^x\sin\left(x\right)\right)$
12

The integral results in

$\frac{1}{2}\left(e^x\cos\left(x\right)+e^x\sin\left(x\right)\right)$
13

Gather the results of all integrals

$\frac{1}{2}\left(e^x\cos\left(x\right)+e^x\sin\left(x\right)\right)$
14

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{1}{2}\left(e^x\cos\left(x\right)+e^x\sin\left(x\right)\right)+C_0$
15

Expand and simplify

$\frac{1}{2}e^x\cos\left(x\right)+\frac{1}{2}e^x\sin\left(x\right)+C_0$

Final answer to the problem

$\frac{1}{2}e^x\cos\left(x\right)+\frac{1}{2}e^x\sin\left(x\right)+C_0$

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Function Plot

Plotting: $\frac{1}{2}e^x\cos\left(x\right)+\frac{1}{2}e^x\sin\left(x\right)+C_0$

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7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

How to improve your answer:

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Main Topic: Integrals of Exponential Functions

Those are integrals that involve exponential functions. Recall that an exponential function is a function of the form f(x)=a^x.

Used Formulas

See formulas (5)

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