Math / Calculus

QuestionВычислить неопределенные интегралы, используя метод интегрирования по частям:
1. $\int 2 x \cos x d x$

Studdy Solution

STEP 1

1. The problem requires the use of integration by parts.
2. The formula for integration by parts is $\int u \, dv = uv - \int v \, du$ .
3. We need to choose $u$ and $dv$ appropriately from the integrand.

STEP 2

1. Identify $u$ and $dv$ .
2. Differentiate $u$ to find $du$ .
3. Integrate $dv$ to find $v$ .
4. Apply the integration by parts formula.
5. Simplify and solve the resulting integral.

STEP 3

Identify $u$ and $dv$ from the integral $\int 2x \cos x \, dx$ .
Let: $u = 2x$ $dv = \cos x \, dx$

STEP 4

Differentiate $u$ to find $du$ :
$du = \frac{d}{dx}(2x) \, dx = 2 \, dx$

STEP 5

Integrate $dv$ to find $v$ :
$v = \int \cos x \, dx = \sin x$

STEP 6

Apply the integration by parts formula $\int u \, dv = uv - \int v \, du$ :
$\int 2x \cos x \, dx = 2x \sin x - \int \sin x \cdot 2 \, dx$

STEP 7

Simplify the expression:
$= 2x \sin x - 2 \int \sin x \, dx$

STEP 8

Solve the remaining integral $\int \sin x \, dx$ :
$\int \sin x \, dx = -\cos x$
Substitute back into the expression:
$2x \sin x - 2(-\cos x)$
$= 2x \sin x + 2\cos x + C$
where $C$ is the constant of integration.
The indefinite integral is:
$\boxed{2x \sin x + 2\cos x + C}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms