What is the integral of (x+1)5^{(x+1)^2} ?

The integral of (x+1)5^{(x+1)^2} is (5^{(x+1)^2})/(2ln(5))+C

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integral of (x+1)5^{(x+1)^2}

\int (x + 1) 5^{(x + 1)^{2}} d x

\frac{5 ^{(x + 1)^{2}}}{2 ln ( 5 )} + C

Solution steps

\int (x + 1) \cdot 5^{(x + 1)^{2}} d x

Apply u-substitution

: \int \frac{5 ^{u}}{2} d u

= \int \frac{5 ^{u}}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int 5^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

= \frac{1}{2} \cdot \frac{5 ^{u}}{ln ( 5 )}

Substitute back

u = (x + 1)^{2}

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

Simplify

\frac{1}{2} \cdot \frac{5 ^{(x + 1)^{2}}}{ln ( 5 )} : \frac{5 ^{(x + 1)^{2}}}{2 ln ( 5 )}

= \frac{5 ^{(x + 1)^{2}}}{2 ln ( 5 )}

Add a constant to the solution

= \frac{5 ^{(x + 1)^{2}}}{2 ln ( 5 )} + C

\frac{d}{d x} (e^{x} - 3 x^{2})

\frac{d}{d x} (25 + x^{2})

x y^{2} - y^{^{'}} x^{2} = 0

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

x \to 0 lim (\frac{e ^{3 x} - e ^{- 3 x}}{4 x})

What is the integral of (x+1)5^{(x+1)^2} ?
The integral of (x+1)5^{(x+1)^2} is (5^{(x+1)^2})/(2ln(5))+C