What is the integral of (ln(t))/(tln(t^2)) ?

The integral of (ln(t))/(tln(t^2)) is 1/2 ln(t)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (ln(t))/(tln(t^2))

\int \frac{ln ( t )}{t ln ( t ^{2} )} d t

\frac{1}{2} ln (t) + C

Solution steps

\int \frac{ln ( t )}{t ln ( t ^{2} )} d t

\frac{ln ( t )}{t ln ( t ^{2} )} = \frac{ln ( t ) - t ln ( t ^{2} )}{t ln ( t ^{2} )} + 1

= \int \frac{ln ( t ) - t ln ( t ^{2} )}{t ln ( t ^{2} )} + 1 d t

Apply u-substitution

: \int \frac{1 - 2 e ^{u}}{2} + e^{u} d u

= \int \frac{1 - 2 e ^{u}}{2} + e^{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1 - 2 e ^{u}}{2} d u + \int e^{u} d u

\int \frac{1 - 2 e ^{u}}{2} d u = \frac{1}{2} u - e^{u}

\int e^{u} d u = e^{u}

= \frac{1}{2} u - e^{u} + e^{u}

Substitute back

u = ln (t)

= \frac{1}{2} ln (t) - e^{l n (t)} + e^{l n (t)}

Simplify

= \frac{1}{2} ln (t)

Add a constant to the solution

= \frac{1}{2} ln (t) + C

\int \frac{u ^{2}}{2 1 - u ^{2}} d u

\int 3 π cos (π) x d x

\int e^{- t (s - 1)} d t

\int (3 x^{2} + 2 x - 5) d x

\int \frac{x}{x ( x + 2 )} d x

What is the integral of (ln(t))/(tln(t^2)) ?
The integral of (ln(t))/(tln(t^2)) is 1/2 ln(t)+C