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Integrate the function $\cos\left(x\right)$ from 0 to $\frac{\pi }{2}$

Step-by-step Solution

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Solution

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Final Answer

$-x\cos\left(x\right)+1$
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Step-by-step Solution

Specify the solving method

1

Simplifying

$\int_{0}^{\frac{\pi}{2}}\cos\left(x\right)dx$
2

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

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
3

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=\cos\left(x\right)}\\ \displaystyle{du=-\sin\left(x\right)dx}\end{matrix}$
4

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=1dx}\\ \displaystyle{\int dv=\int 1dx}\end{matrix}$
5

Solve the integral

$v=\int1dx$
6

The integral of a constant is equal to the constant times the integral's variable

$x$
7

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

$\left[x\cos\left(x\right)\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}} x\sin\left(x\right)dx$
8

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

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
9

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x}\\ \displaystyle{du=dx}\end{matrix}$
10

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\sin\left(x\right)dx}\\ \displaystyle{\int dv=\int \sin\left(x\right)dx}\end{matrix}$
11

Solve the integral

$v=\int\sin\left(x\right)dx$
12

Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$

$-\cos\left(x\right)$
13

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

$\left[x\cos\left(x\right)\right]_{0}^{\frac{\pi}{2}}-x\cos\left(x\right)-\int_{0}^{\frac{\pi}{2}}-\cos\left(x\right)dx$
14

The integral $\int_{0}^{\frac{\pi}{2}}\cos\left(x\right)dx$ results in: $1$

$1$
15

Gather the results of all integrals

$\left[x\cos\left(x\right)\right]_{0}^{\frac{\pi}{2}}-x\cos\left(x\right)+1$
16

Evaluate the definite integral

$\frac{\pi}{2}\cos\left(\frac{\pi}{2}\right)- 0\cos\left(0\right)-x\cos\left(x\right)+1$
17

Simplify the expression inside the integral

$-x\cos\left(x\right)+1$

Final Answer

$-x\cos\left(x\right)+1$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Solve integral of cosxdx from 0 to \pi /2 using basic integralsSolve integral of cosxdx from 0 to \pi /2 using u-substitutionSolve integral of cosxdx from 0 to \pi /2 using tabular integrationSolve integral of cosxdx from 0 to \pi /2 using weierstrass substitution

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Function Plot

Plotting: $\cos\left(x\right)$

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(◻)
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◻/◻
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◻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: Definite Integrals

Given a function f(x) and the interval [a,b], the definite integral is equal to the area that is bounded by the graph of f(x), the x-axis and the vertical lines x=a and x=b

Used Formulas

6. See formulas

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