What is the integral of 4e^x(e^x+1)^3 ?

The integral of 4e^x(e^x+1)^3 is (e^x+1)^4+C

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integral of 4e^x(e^x+1)^3

\int 4 e^{x} (e^{x} + 1)^{3} d x

(e^{x} + 1)^{4} + C

Solution steps

\int 4 e^{x} (e^{x} + 1)^{3} d x

Take the constant out:

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

= 4 \cdot \int e^{x} (e^{x} + 1)^{3} d x

Apply u-substitution

= 4 \cdot \int u^{3} d u

Apply the Power Rule

= 4 \cdot \frac{u ^{4}}{4}

Substitute back

u = e^{x} + 1

= 4 \cdot \frac{( e ^{x} + 1 ) ^{4}}{4}

Simplify

4 \cdot \frac{( e ^{x} + 1 ) ^{4}}{4} : (e^{x} + 1)^{4}

= (e^{x} + 1)^{4}

Add a constant to the solution

= (e^{x} + 1)^{4} + C

What is the integral of 4e^x(e^x+1)^3 ?
The integral of 4e^x(e^x+1)^3 is (e^x+1)^4+C