What is the integral of x(4x^2+5)^7 ?

The integral of x(4x^2+5)^7 is 1/64 (4x^2+5)^8+C

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integral of x(4x^2+5)^7

\int x (4 x^{2} + 5)^{7} d x

\frac{1}{64} (4 x^{2} + 5)^{8} + C

Solution steps

\int x (4 x^{2} + 5)^{7} d x

Apply u-substitution

: \int \frac{u ^{7}}{8} d u

= \int \frac{u ^{7}}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= \frac{1}{8} \cdot \frac{u ^{8}}{8}

Substitute back

u = 4 x^{2} + 5

= \frac{1}{8} \cdot \frac{( 4 x ^{2} + 5 ) ^{8}}{8}

Simplify

\frac{1}{8} \cdot \frac{( 4 x ^{2} + 5 ) ^{8}}{8} : \frac{1}{64} (4 x^{2} + 5)^{8}

= \frac{1}{64} (4 x^{2} + 5)^{8}

Add a constant to the solution

= \frac{1}{64} (4 x^{2} + 5)^{8} + C

f (t) = e^{t}

\frac{12}{( 2 sin ( x ) + 6 cos ( x ) ) ^{2}}

\int_{2}^{3} \frac{11}{3 - x} d x

y^{^{''}} - 6 y^{^{'}} + 9 y = t, y (0) = 0, y^{^{'}} (0) = 1

\frac{d}{d x} (\frac{1}{x} - \frac{1}{sin ( x )})

What is the integral of x(4x^2+5)^7 ?
The integral of x(4x^2+5)^7 is 1/64 (4x^2+5)^8+C