What is the integral of 4t^4e^{2t^5} ?

The integral of 4t^4e^{2t^5} is 2/5 e^{2t^5}+C

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integral of 4t^4e^{2t^5}

\int 4 t^{4} e^{2 t^{5}} d t

\frac{2}{5} e^{2 t^{5}} + C

Solution steps

\int 4 t^{4} e^{2 t^{5}} d t

Take the constant out:

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

= 4 \cdot \int t^{4} e^{2 t^{5}} d t

Apply u-substitution

: \int \frac{e ^{2 u^{\frac{5}{4}}} u ^{\frac{1}{4}}}{4} d u

= 4 \cdot \int \frac{e ^{2 u^{\frac{5}{4}}} u ^{\frac{1}{4}}}{4} d u

Take the constant out:

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

= 4 \cdot \frac{1}{4} \cdot \int e^{2 u^{\frac{5}{4}}} u^{\frac{1}{4}} d u

Apply u-substitution

: \int \frac{4 e ^{2 v}}{5} d v

= 4 \cdot \frac{1}{4} \cdot \int \frac{4 e ^{2 v}}{5} d v

Take the constant out:

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

= 4 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \int e^{2 v} d v

Apply u-substitution

: \int e^{w} \frac{1}{2} d w

= 4 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \int e^{w} \frac{1}{2} d w

Take the constant out:

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

= 4 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{2} \cdot \int e^{w} d w

Use the common integral:

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

= 4 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{2} e^{w}

Substitute back

= 4 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{2} e^{2 (t^{4})^{\frac{5}{4}}}

Simplify

4 \cdot \frac{1}{4} \cdot \frac{4}{5} \cdot \frac{1}{2} e^{2 (t^{4})^{\frac{5}{4}}} : \frac{2}{5} e^{2 t^{5}}

= \frac{2}{5} e^{2 t^{5}}

Add a constant to the solution

= \frac{2}{5} e^{2 t^{5}} + C

y^{^{''}} + 4 y = cos (2 x)

\int x^{2} arctan (3 x) d x

\frac{\partial}{\partial x} (4 x + 5)

f (x) = x^{2} - 3

x^{2} y^{^{'}} = y - x y

What is the integral of 4t^4e^{2t^5} ?
The integral of 4t^4e^{2t^5} is 2/5 e^{2t^5}+C