What is the integral of (5t+1)^{100} ?

The integral of (5t+1)^{100} is 1/505 (5t+1)^{101}+C

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integral of (5t+1)^{100}

\int (5 t + 1)^{100} d t

\frac{1}{505} (5 t + 1)^{101} + C

Solution steps

\int (5 t + 1)^{100} d t

Apply u-substitution

: \int \frac{u ^{100}}{5} d u

= \int \frac{u ^{100}}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int u^{100} d u

Apply the Power Rule

: \frac{u ^{101}}{101}

= \frac{1}{5} \cdot \frac{u ^{101}}{101}

Substitute back

u = 5 t + 1

= \frac{1}{5} \cdot \frac{( 5 t + 1 ) ^{101}}{101}

Simplify

\frac{1}{5} \cdot \frac{( 5 t + 1 ) ^{101}}{101} : \frac{1}{505} (5 t + 1)^{101}

= \frac{1}{505} (5 t + 1)^{101}

Add a constant to the solution

= \frac{1}{505} (5 t + 1)^{101} + C

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What is the integral of (5t+1)^{100} ?
The integral of (5t+1)^{100} is 1/505 (5t+1)^{101}+C