What is the integral of e^{12x}(12) ?

The integral of e^{12x}(12) is e^{12x}+C

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integral of e^{12x}(12)

\int e^{12 x} (12) d x

e^{12 x} + C

Solution steps

\int e^{12 x} \cdot 12 d x

Take the constant out:

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

= 12 \cdot \int e^{12 x} d x

Apply u-substitution

: \int e^{u} \frac{1}{12} d u

= 12 \cdot \int e^{u} \frac{1}{12} d u

Take the constant out:

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

= 12 \cdot \frac{1}{12} \cdot \int e^{u} d u

Use the common integral:

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

= 12 \cdot \frac{1}{12} e^{u}

Substitute back

u = 12 x

= 12 \cdot \frac{1}{12} e^{12 x}

Simplify

12 \cdot \frac{1}{12} e^{12 x} : e^{12 x}

= e^{12 x}

Add a constant to the solution

= e^{12 x} + C

f (x) = 3 x^{2} cos (x) + 8 x

x \to 0 + lim ((sin (x))^{x})

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