Answer:
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx = (e^(-8x))/(2) \bigg( (23)/(8) - x \bigg) + C](https://img.qammunity.org/2021/formulas/mathematics/college/gmxeyy2ntqpf8onwsgjxabo50h3nq8qt8p.png)
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]:
![\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)](https://img.qammunity.org/2021/formulas/mathematics/college/bz16ipe6p14y3f6abzxt2zy0j41tg530u9.png)
Derivative Property [Addition/Subtraction]:
![\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]](https://img.qammunity.org/2021/formulas/mathematics/college/kqosumt4896r7x44jgtw0o7kk6g4d3irvr.png)
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Integration
- Integrals
- [Indefinite Integrals] Integration Constant C
Integration Property [Multiplied Constant]:
![\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx](https://img.qammunity.org/2021/formulas/mathematics/college/kyhrzhajthfkoabkn5u9i412baa68ie7zm.png)
U-Substitution
Integration by Parts:
![\displaystyle \int {u} \, dv = uv - \int {v} \, du](https://img.qammunity.org/2021/formulas/mathematics/college/babomk9eltny0rfoifpt2pbc8iqonzv2j3.png)
- [IBP] LIPET: Logs, inverses, Polynomials, Exponentials, Trig
Explanation:
Step 1: Define
Identify
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx](https://img.qammunity.org/2021/formulas/mathematics/college/1hzglix6876bpcpzyayzj14d3xub8alyth.png)
Step 2: Integrate Pt. 1
- [Integrand] Rewrite [Factor]:
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx = \int {4(x - 3)e^(-8x)} \, dx](https://img.qammunity.org/2021/formulas/mathematics/college/im4cnqnsxfunpeh09j9lrr0jynexgnh98y.png)
- [Integral] Rewrite [Integration Property - Multiplied Constant]:
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx = 4 \int {(x - 3)e^(-8x)} \, dx](https://img.qammunity.org/2021/formulas/mathematics/college/m5s51coezfgyhh7x1rdffzsakhfdorc1nw.png)
Step 3: Integrate Pt. 2
Identify variables for integration by parts using LIPET.
- Set u:
![\displaystyle u = x - 3](https://img.qammunity.org/2021/formulas/mathematics/college/4qq2yscn909re1strgd3tv7xq4o0cjzq2t.png)
- [u] Basic Power Rule [Derivative Property - Addition/Subtraction]:
![\displaystyle du = dx](https://img.qammunity.org/2021/formulas/mathematics/college/gmxgfw80hw9mgrchypwdjrxlflzdlw94ze.png)
- Set dv:
![\displaystyle dv = e^(-8x) \ dx](https://img.qammunity.org/2021/formulas/mathematics/college/t24tovr19bom73f3e6rni3awf1ubjiuw3x.png)
- [dv] Exponential Integration [U-Substitution]:
![\displaystyle v = (-e^(-8x))/(8)](https://img.qammunity.org/2021/formulas/mathematics/college/dyby7o5qkpnyu7ly25rxnhaszwuqjl6ao4.png)
Step 4: Integrate Pt. 3
- [Integral] Integration by Parts:
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx = 4 \bigg( (-(x - 3)e^(-8x))/(8) - \int {(-e^(-8x))/(8)} \, dx \bigg)](https://img.qammunity.org/2021/formulas/mathematics/college/1bguu9udzumbdxcvvz0umjn4dp267mpgh5.png)
- [Integral] Rewrite [Integration Property - Multiplied Constant]:
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx = 4 \bigg( (-(x - 3)e^(-8x))/(8) + (1)/(8) \int {e^(-8x)} \, dx \bigg)](https://img.qammunity.org/2021/formulas/mathematics/college/1d66dqsw9tuji7eul8tcnqcto1ty1slxxb.png)
- Factor:
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx = (1)/(2) \bigg( -(x - 3)e^(-8x) + \int {e^(-8x)} \, dx \bigg)](https://img.qammunity.org/2021/formulas/mathematics/college/otdri9bqowbp7hrzee4kw0xd6v28kluz7p.png)
Step 5: Integrate Pt. 4
Identify variables for u-substitution.
- Set u:
![\displaystyle u = -8x](https://img.qammunity.org/2021/formulas/mathematics/college/gfcpnvzmfl05hsswm9t3d1wm5cua2sdbon.png)
- [u] Basic Power Rule [Derivative Rule - Multiplied Constant]:
![\displaystyle du = -8 \ dx](https://img.qammunity.org/2021/formulas/mathematics/college/a17mhydfze8gh75eq6rk9xjb8jjg1adb2p.png)
Step 6: Integrate Pt. 5
- [Integral] Rewrite [Integration Property - Multiplied Constant]:
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx = (1)/(2) \bigg( -(x - 3)e^(-8x) - (1)/(8) \int {-8e^(-8x)} \, dx \bigg)](https://img.qammunity.org/2021/formulas/mathematics/college/w7cql0z0xz1nzoz8yat6ekl8xtt0c9554f.png)
- [Integral] U-Substitution:
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx = (1)/(2) \bigg( (x - 3)e^(-8x) - (1)/(8) \int {e^u} \, dx \bigg)](https://img.qammunity.org/2021/formulas/mathematics/college/9bem5xamvrxlqqy5w9q6afe1m2t8si1who.png)
- [Integral] Exponential Integration:
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx = (1)/(2) \bigg( (x - 3)e^(-8x) - (e^u)/(8) \bigg) + C](https://img.qammunity.org/2021/formulas/mathematics/college/zfj1ntl34ou4yq8pggtrw08fbx3t7wpej4.png)
- [u] Back-Substitute:
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx = (1)/(2) \bigg( (x - 3)e^(-8x) - (e^(-8x))/(8) \bigg) + C](https://img.qammunity.org/2021/formulas/mathematics/college/ney7q7yd1h9iylcdzyx4cdt5ry9e7kzh5d.png)
- Factor:
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx = (e^(-8x))/(2) \bigg( -(x - 3) - (1)/(8) \bigg) + C](https://img.qammunity.org/2021/formulas/mathematics/college/rjvuh5nx0iwl49t0n7wvodekw2pv9xxicy.png)
- Simplify:
![\displaystyle \int {(4x - 12)e^(-8x)} \, dx = (e^(-8x))/(2) \bigg( (23)/(8) - x \bigg) + C](https://img.qammunity.org/2021/formulas/mathematics/college/gmxeyy2ntqpf8onwsgjxabo50h3nq8qt8p.png)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration