What is the integral of (4t^4+48t^9)/(t^5+6t^{10)} ?

The integral of (4t^4+48t^9)/(t^5+6t^{10)} is 4/5 ln|t^5+6t^{10}|+C

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integral of (4t^4+48t^9)/(t^5+6t^{10)}

\int \frac{4 t ^{4} + 48 t ^{9}}{t ^{5} + 6 t ^{10}} d t

\frac{4}{5} ln t^{5} + 6 t^{10} + C

Solution steps

\int \frac{4 t ^{4} + 48 t ^{9}}{t ^{5} + 6 t ^{10}} d t

Apply u-substitution

= \int \frac{4}{5 u} d u

Take the constant out:

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

= \frac{4}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{4}{5} ln ∣ u ∣

Substitute back

u = t^{5} + 6 t^{10}

= \frac{4}{5} ln t^{5} + 6 t^{10}

Add a constant to the solution

= \frac{4}{5} ln t^{5} + 6 t^{10} + C

What is the integral of (4t^4+48t^9)/(t^5+6t^{10)} ?
The integral of (4t^4+48t^9)/(t^5+6t^{10)} is 4/5 ln|t^5+6t^{10}|+C