What is the integral of x^{n-1}sqrt(a+bx^n) ?

The integral of x^{n-1}sqrt(a+bx^n) is (2(a+bx^n)^{3/2})/(3bn)+C

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integral of x^{n-1}sqrt(a+bx^n)

\int x^{n - 1} a + b x^{n} d x

\frac{2 ( a + b x ^{n} ) ^{\frac{3}{2}}}{3 bn} + C

Solution steps

\int x^{n - 1} a + b x^{n} d x

Apply u-substitution

: \int \frac{u}{bn} d u

= \int \frac{u}{bn} d u

Take the constant out:

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

= \frac{1}{bn} \cdot \int u d u

Apply the Power Rule

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

= \frac{1}{bn} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = a + b x^{n}

= \frac{1}{bn} \cdot \frac{2}{3} (a + b x^{n})^{\frac{3}{2}}

Simplify

\frac{1}{bn} \cdot \frac{2}{3} (a + b x^{n})^{\frac{3}{2}} : \frac{2 ( a + b x ^{n} ) ^{\frac{3}{2}}}{3 bn}

= \frac{2 ( a + b x ^{n} ) ^{\frac{3}{2}}}{3 bn}

Add a constant to the solution

= \frac{2 ( a + b x ^{n} ) ^{\frac{3}{2}}}{3 bn} + C

\frac{d}{d x} (x (x + 4)^{2})

\frac{\partial}{\partial x} (10 x^{2} y + 5 x y^{3})

\frac{d}{d x} ((ln (x))^{8})

y^{^{''}} + 4 y^{^{'}} + 4 y = \frac{( e ^{2 x} )}{( x )}

\frac{A}{y ^{8}} + B e^{y}

What is the integral of x^{n-1}sqrt(a+bx^n) ?
The integral of x^{n-1}sqrt(a+bx^n) is (2(a+bx^n)^{3/2})/(3bn)+C