What is the integral of sqrt(t)sqrt(1+t\sqrt{t)} ?

The integral of sqrt(t)sqrt(1+t\sqrt{t)} is 4/9 (1+tsqrt(t))^{3/2}+C

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integral of sqrt(t)sqrt(1+t\sqrt{t)}

\int t 1 + t t d t

\frac{4}{9} (1 + t t)^{\frac{3}{2}} + C

Solution steps

\int t 1 + t t d t

Apply u-substitution

: \int \frac{2 u}{3} d u

= \int \frac{2 u}{3} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{2}{3} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + t t

= \frac{2}{3} \cdot \frac{2}{3} (1 + t t)^{\frac{3}{2}}

Simplify

\frac{2}{3} \cdot \frac{2}{3} (1 + t t)^{\frac{3}{2}} : \frac{4}{9} (1 + t t)^{\frac{3}{2}}

= \frac{4}{9} (1 + t t)^{\frac{3}{2}}

Add a constant to the solution

= \frac{4}{9} (1 + t t)^{\frac{3}{2}} + C

f (x) = 1 - 10 x, at x = - 1

\int (- 1)^{n} x^{2 n} d x

x \to 0 lim ((x - 2)^{2})

\frac{d ^{2}}{d x ^{2}} (x e^{x})

\frac{d y}{d t} + 2 y = 0

What is the integral of sqrt(t)sqrt(1+t\sqrt{t)} ?
The integral of sqrt(t)sqrt(1+t\sqrt{t)} is 4/9 (1+tsqrt(t))^{3/2}+C