What is the integral of 3m^2(m^3+4)^{53} ?

The integral of 3m^2(m^3+4)^{53} is 1/54 (m^3+4)^{54}+C

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integral of 3m^2(m^3+4)^{53}

\int 3 m^{2} (m^{3} + 4)^{53} d m

\frac{1}{54} (m^{3} + 4)^{54} + C

Solution steps

\int 3 m^{2} (m^{3} + 4)^{53} d m

Take the constant out:

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

= 3 \cdot \int m^{2} (m^{3} + 4)^{53} d m

Apply u-substitution

: \int \frac{u ^{53}}{3} d u

= 3 \cdot \int \frac{u ^{53}}{3} d u

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \int u^{53} d u

Apply the Power Rule

: \frac{u ^{54}}{54}

= 3 \cdot \frac{1}{3} \cdot \frac{u ^{54}}{54}

Substitute back

u = m^{3} + 4

= 3 \cdot \frac{1}{3} \cdot \frac{( m ^{3} + 4 ) ^{54}}{54}

Simplify

3 \cdot \frac{1}{3} \cdot \frac{( m ^{3} + 4 ) ^{54}}{54} : \frac{1}{54} (m^{3} + 4)^{54}

= \frac{1}{54} (m^{3} + 4)^{54}

Add a constant to the solution

= \frac{1}{54} (m^{3} + 4)^{54} + C

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What is the integral of 3m^2(m^3+4)^{53} ?
The integral of 3m^2(m^3+4)^{53} is 1/54 (m^3+4)^{54}+C