What is the integral of (4t)/(sqrt(3t^2-7)) ?

The integral of (4t)/(sqrt(3t^2-7)) is 4/3 sqrt(3t^2-7)+C

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integral of (4t)/(sqrt(3t^2-7))

\int \frac{4 t}{3 t ^{2} - 7} d t

\frac{4}{3} 3 t^{2} - 7 + C

Solution steps

\int \frac{4 t}{3 t ^{2} - 7} d t

Take the constant out:

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

= 4 \cdot \int \frac{t}{3 t ^{2} - 7} d t

Apply u-substitution

: \int \frac{1}{6 u} d u

= 4 \cdot \int \frac{1}{6 u} d u

Take the constant out:

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

= 4 \cdot \frac{1}{6} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= 4 \cdot \frac{1}{6} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 3 t^{2} - 7

= 4 \cdot \frac{1}{6} \cdot 2 (3 t^{2} - 7)^{\frac{1}{2}}

Simplify

4 \cdot \frac{1}{6} \cdot 2 (3 t^{2} - 7)^{\frac{1}{2}} : \frac{4}{3} 3 t^{2} - 7

= \frac{4}{3} 3 t^{2} - 7

Add a constant to the solution

= \frac{4}{3} 3 t^{2} - 7 + C

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What is the integral of (4t)/(sqrt(3t^2-7)) ?
The integral of (4t)/(sqrt(3t^2-7)) is 4/3 sqrt(3t^2-7)+C