What is the integral of tcos(npit) ?

The integral of tcos(npit) is 1/(pi^2n^2)(pintsin(pint)+cos(pint))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of tcos(npit)

\int t cos (nπ t) d t

\frac{1}{π ^{2} n ^{2}} (πn t sin (πn t) + cos (πn t)) + C

Solution steps

\int t cos (nπ t) d t

Apply u-substitution

: \int \frac{u cos ( u )}{π ^{2} n ^{2}} d u

= \int \frac{u cos ( u )}{π ^{2} n ^{2}} d u

Take the constant out:

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

= \frac{1}{π ^{2} n ^{2}} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= \frac{1}{π ^{2} n ^{2}} (u sin (u) - \int sin (u) d u)

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

= \frac{1}{π ^{2} n ^{2}} (u sin (u) - (- cos (u)))

Substitute back

u = n t π

= \frac{1}{π ^{2} n ^{2}} (n t π sin (n t π) - (- cos (n t π)))

Simplify

= \frac{1}{π ^{2} n ^{2}} (πn t sin (πn t) + cos (πn t))

Add a constant to the solution

= \frac{1}{π ^{2} n ^{2}} (πn t sin (πn t) + cos (πn t)) + C

\int \frac{3 x ^{2} - 4}{x ^{2} - 1} d x

\int sec (21 x) tan (21 x) d x

\int \frac{x + 6}{x ^{2} ( x ^{2} + 3 )} d x

\int e^{t a n (24 x)} sec^{2} (24 x) d x

\int \frac{3 x ^{2} + 1}{x ^{2} + 4} d x

What is the integral of tcos(npit) ?
The integral of tcos(npit) is 1/(pi^2n^2)(pintsin(pint)+cos(pint))+C