What is the integral of (x^2+1)(x^3+3x)^4 ?

The integral of (x^2+1)(x^3+3x)^4 is 1/15 (x^3+3x)^5+C

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integral of (x^2+1)(x^3+3x)^4

\int (x^{2} + 1) (x^{3} + 3 x)^{4} d x

\frac{1}{15} (x^{3} + 3 x)^{5} + C

Solution steps

\int (x^{2} + 1) (x^{3} + 3 x)^{4} d x

Apply u-substitution

: \int \frac{u ^{4}}{3} d u

= \int \frac{u ^{4}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= \frac{1}{3} \cdot \frac{u ^{5}}{5}

Substitute back

u = x^{3} + 3 x

= \frac{1}{3} \cdot \frac{( x ^{3} + 3 x ) ^{5}}{5}

Simplify

\frac{1}{3} \cdot \frac{( x ^{3} + 3 x ) ^{5}}{5} : \frac{1}{15} (x^{3} + 3 x)^{5}

= \frac{1}{15} (x^{3} + 3 x)^{5}

Add a constant to the solution

= \frac{1}{15} (x^{3} + 3 x)^{5} + C

\int \frac{2 x + 5}{x ^{2} + 5 x} d x

f (x) = 7 + 5 x

y^{^{'}} = k \cdot y

\frac{( 3 t - 1 )}{t - 2}

y = \frac{4 x}{x + 2}

What is the integral of (x^2+1)(x^3+3x)^4 ?
The integral of (x^2+1)(x^3+3x)^4 is 1/15 (x^3+3x)^5+C