What is the integral of 5(400+5t)^3 ?

The integral of 5(400+5t)^3 is 1/4 (400+5t)^4+C

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integral of 5(400+5t)^3

\int 5 (400 + 5 t)^{3} d t

\frac{1}{4} (400 + 5 t)^{4} + C

Solution steps

\int 5 (400 + 5 t)^{3} d t

Take the constant out:

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

= 5 \cdot \int (400 + 5 t)^{3} d t

Apply u-substitution

: \int \frac{u ^{3}}{5} d u

= 5 \cdot \int \frac{u ^{3}}{5} d u

Take the constant out:

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

= 5 \cdot \frac{1}{5} \cdot \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= 5 \cdot \frac{1}{5} \cdot \frac{u ^{4}}{4}

Substitute back

u = 400 + 5 t

= 5 \cdot \frac{1}{5} \cdot \frac{( 400 + 5 t ) ^{4}}{4}

Simplify

5 \cdot \frac{1}{5} \cdot \frac{( 400 + 5 t ) ^{4}}{4} : \frac{1}{4} (400 + 5 t)^{4}

= \frac{1}{4} (400 + 5 t)^{4}

Add a constant to the solution

= \frac{1}{4} (400 + 5 t)^{4} + C

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What is the integral of 5(400+5t)^3 ?
The integral of 5(400+5t)^3 is 1/4 (400+5t)^4+C