Evaluate the integral by using a substitution prior to integration by parts.
-$$\int \frac { 1 } { 2 } \mathrm { e }^{ \sqrt { 7 x + 8 }}\mathrm { dx }$$
A) $( 7 x + 8 ) e^{ \sqrt { 7 x + 8 } }+ C$
B) $\frac { 1 } { 7 } \mathrm { e }^{ \sqrt { 7 x + 8 }} [ \sqrt { 7 x + 8 } - 7 ] + C$
C) $\frac { \sqrt { 7 x + 8 } } { 7 }\mathrm { e }^{ \sqrt { 7 x + 8 } }+ C$
D) $\frac { 1 } { 7 } e^{ \sqrt { 7 x + 8 } }[ \sqrt { 7 x + 8 } - 1 ] + C$

Question

Quizplus · Accepted Answer

First, use the substitution $ u = \sqrt{7x + 8} $, then $ du = \frac{7}{2u} dx $. Solving for $ dx $, we get $ dx = \frac{2u}{7} du $. Substitute into the integral to get $\int \frac{1}{2} e^u \cdot \frac{2u}{7} du = \frac{1}{7} \int u e^u du$. Use integration by parts with $ v = e^u $ and $ dw = u du $, leading to the result $\frac{1}{7} e^u (u - 1) + C$. Substituting back $ u = \sqrt{7x + 8} $ gives the answer.

Evaluate the Integral by Using a Substitution Prior to Integration ∫12e7x+8dx\int \frac { 1 } { 2 } \mathrm { e }^{ \sqrt { 7 x + 8 }}\mathrm { dx }∫21​e7x+8​dx

Evaluate the Integral by Using a Substitution Prior to Integration $\int \frac { 1 } { 2 } \mathrm { e }^{ \sqrt { 7 x + 8 }}\mathrm { dx }$