Integral Calculus Question 427
Question: To evaluate $ \int_{{}}^{{}}{x^{3}{e^{3x^{2}+5}}}dx $ , the simplest way is to
Options:
A) Substitute $ x^{2}=t $
B) Substitute $ (3x^{2}+5)=t $
C) Integrate by parts
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
$ \int_{{}}^{{}}{x^{3}{e^{3x^{2}+5}}dx} $ The simplest way is substituting $ (3x^{2}+5)=t. $ Put $ t=3x^{2}+5\Rightarrow dx=\frac{dt}{6x}, $ then $ \int_{{}}^{{}}{x^{3}{e^{3x^{2}+5}}dx}=\frac{1}{6}\int_{{}}^{{}}{( \frac{t-5}{3} )e^{t}dt} $ $ =\frac{1}{18}\int_{{}}^{{}}{[te^{t}-5e^{t}]dt}=\frac{1}{18}\int_{{}}^{{}}{te^{t}dt}-\frac{5}{18}\int_{{}}^{{}}{e^{t}dt} $ $ =\frac{1}{18}[ te^{t}-\int_{{}}^{{}}{e^{t}dt} ]-\frac{5}{18}\int_{{}}^{{}}{e^{t}dt}+c $ $ =\frac{1}{18}(te^{t})-\frac{1}{3}e^{t}+c $ $ =\frac{1}{18}(3x^{2}+5),{e^{3x^{2}+5}}-\frac{1}{3}{e^{3x^{2}+5}}+c. $
BETA
