What is the integral of (1+2x)^42 ?

The integral of (1+2x)^42 is 1/5 (1+2x)^5+C

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integral of (1+2x)^42

\int (1 + 2 x)^{4} 2 d x

\frac{1}{5} (1 + 2 x)^{5} + C

Solution steps

\int (1 + 2 x)^{4} \cdot 2 d x

Take the constant out:

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

= 2 \cdot \int (1 + 2 x)^{4} d x

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

= 2 \cdot \frac{1}{2} \cdot \frac{u ^{5}}{5}

Substitute back

u = 1 + 2 x

= 2 \cdot \frac{1}{2} \cdot \frac{( 1 + 2 x ) ^{5}}{5}

Simplify

2 \cdot \frac{1}{2} \cdot \frac{( 1 + 2 x ) ^{5}}{5} : \frac{1}{5} (1 + 2 x)^{5}

= \frac{1}{5} (1 + 2 x)^{5}

Add a constant to the solution

= \frac{1}{5} (1 + 2 x)^{5} + C