What is the integral from 0 to pi of x^2*cos(n*x) ?

The integral from 0 to pi of x^2*cos(n*x) is (2pi(-1)^n)/(n^2)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to pi of x^2cos(nx)

\int_{0}^{π} x^{2} \cdot cos (n \cdot x) d x

\frac{2 π ( - 1 ) ^{n}}{n ^{2}}

Solution steps

\int_{0}^{π} x^{2} cos (n x) d x

Apply u-substitution

: \int_{0}^{nπ} \frac{u ^{2} cos ( u )}{n ^{3}} d u

= \int_{0}^{nπ} \frac{u ^{2} cos ( u )}{n ^{3}} d u

Take the constant out:

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

= \frac{1}{n ^{3}} \cdot \int_{0}^{nπ} u^{2} cos (u) d u

Apply Integration By Parts

: [u^{2} sin (u) - \int 2 u sin (u) d u]_{0}^{nπ}

= \frac{1}{n ^{3}} [u^{2} sin (u) - \int 2 u sin (u) d u]_{0}^{nπ}

\int 2 u sin (u) d u = 2 (- u cos (u) + sin (u))

= \frac{1}{n ^{3}} [u^{2} sin (u) - 2 (- u cos (u) + sin (u))]_{0}^{nπ}

Compute the boundaries:

2 π (- 1)^{n} n

= \frac{1}{n ^{3}} \cdot 2 π (- 1)^{n} n

Simplify

= \frac{2 π ( - 1 ) ^{n}}{n ^{2}}

\int_{0}^{2} x^{2} e^{3 x} d x

\int_{- x}^{x} cos (t^{2} + 1) d t

\int_{1}^{2} \frac{3}{5 x} d x

\int_{1}^{2} (1 - 2 x^{2}) d x

\int_{0}^{π} 2 x cos (x^{2}) d x

What is the integral from 0 to pi of x^2*cos(n*x) ?
The integral from 0 to pi of x^2*cos(n*x) is (2pi(-1)^n)/(n^2)