Evaluate the integral using integration by parts.
-$\int x ^ { 2 }$ $\mathrm { e } ^ { 2 x }$ dx
A) $\mathrm { e } ^ { 2 x }$ $\left( \frac { x ^ { 2 } - x } { 2 } \right)$ + 1 + C
B) $\mathrm { e } ^ { 2 x }$ $\left( \frac { x ^ { 2 } - 4 x } { 2 } \right)$ + C
C) $\mathrm { e } ^ { 2 x }$ $\left( \frac { x ^ { 2 } - 2 x } { 2 } \right)$ + C
D) $\mathrm { e } ^ { 2 x }$ $\left( \frac { 2 x ^ { 2 } - 2 x + 1 } { 4 } \right)$ + C

Question

Quizplus · Accepted Answer

Integration by parts formula is $\int u dv = uv - \int v du$. Let $u = x^2$ and $dv = e^{2x} dx$, then $du = 2x dx$ and $v = \frac{1}{2}e^{2x}$. Applying the formula, we get $\frac{1}{2}x^2e^{2x} - \int \frac{1}{2} \cdot 2x e^{2x} dx$. The second integral requires another application of integration by parts with $u = x$ and $dv = e^{2x} dx$, resulting in $\frac{1}{2}x^2e^{2x} - \left( \frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x} \right) + C$, which simplifies to $\frac{e^{2x}(2x^2 - 2x + 1)}{4} + C$.

Evaluate the Integral Using Integration by Parts ∫x2\int x ^ { 2 }∫x2

Evaluate the Integral Using Integration by Parts $\int x ^ { 2 }$