What is the integral of 3x^3tanh(x^4) ?

The integral of 3x^3tanh(x^4) is 3/4 ln|cosh(x^4)|+C

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integral of 3x^3tanh(x^4)

\int 3 x^{3} tanh (x^{4}) d x

\frac{3}{4} ln cosh (x^{4}) + C

Solution steps

\int 3 x^{3} tanh (x^{4}) d x

Take the constant out:

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

= 3 \cdot \int x^{3} tanh (x^{4}) d x

Apply u-substitution

: \int \frac{tanh ( u )}{4} d u

= 3 \cdot \int \frac{tanh ( u )}{4} d u

Take the constant out:

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

= 3 \cdot \frac{1}{4} \cdot \int tanh (u) d u

Rewrite using trig identities

= 3 \cdot \frac{1}{4} \cdot \int \frac{sinh ( u )}{cosh ( u )} d u

Apply u-substitution

: \int \frac{1}{v} d v

= 3 \cdot \frac{1}{4} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 3 \cdot \frac{1}{4} ln ∣ v ∣

Substitute back

= 3 \cdot \frac{1}{4} ln cosh (x^{4})

Simplify

3 \cdot \frac{1}{4} ln cosh (x^{4}) : \frac{3}{4} ln cosh (x^{4})

= \frac{3}{4} ln cosh (x^{4})

Add a constant to the solution

= \frac{3}{4} ln cosh (x^{4}) + C

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What is the integral of 3x^3tanh(x^4) ?
The integral of 3x^3tanh(x^4) is 3/4 ln|cosh(x^4)|+C