What is the integral of (t^5)/(\sqrt[7]{at^6+2)} ?

The integral of (t^5)/(\sqrt[7]{at^6+2)} is (7(2+at^6)^{6/7})/(36a)+C

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integral of (t^5)/(\sqrt[7]{at^6+2)}

\frac{7 ( 2 + a t ^{6} ) ^{\frac{6}{7}}}{36 a} + C

Solution steps

Apply u-substitution

Take the constant out:

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

Apply the Power Rule

= \frac{1}{6 a} \cdot \frac{7}{6} u^{\frac{6}{7}}

Substitute back

u = 2 + a t^{6}

= \frac{1}{6 a} \cdot \frac{7}{6} (2 + a t^{6})^{\frac{6}{7}}

Simplify

\frac{1}{6 a} \cdot \frac{7}{6} (2 + a t^{6})^{\frac{6}{7}} : \frac{7 ( 2 + a t ^{6} ) ^{\frac{6}{7}}}{36 a}

= \frac{7 ( 2 + a t ^{6} ) ^{\frac{6}{7}}}{36 a}

Add a constant to the solution

= \frac{7 ( 2 + a t ^{6} ) ^{\frac{6}{7}}}{36 a} + C

What is the integral of (t^5)/(\sqrt[7]{at^6+2)} ?
The integral of (t^5)/(\sqrt[7]{at^6+2)} is (7(2+at^6)^{6/7})/(36a)+C