Substitution rule for definite integral

Calculus Level 2

Evaluate the following definite integral using only the substitution rule and type your answer as a decimal to 4 decimal places:

∫ 1 2 d x ( 3 − 5 x ) 2 \displaystyle \int_{1}^{2} \frac{dx}{(3-5x)^2}


The answer is 0.0714.

Krishna Karthik by Krishna Karthik , 15, Australia

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1 solution

Krishna Karthik
Dec 14, 2018

Let u = 3 − 5 x u=3-5x , then d u = d u d x d x du=\large\frac{du}{dx}dx

d u = − 5 d x du=-5 dx , so d x = − 1 5 d u dx=-\frac{1}{5}du .

To find the new limits of integration: u = 3 − 5 x u=3-5x

u = 3 − 5 ∗ 2 u=3-5*2 (The top limit)

u = 3 − 5 ∗ 1 u=3-5*1 (The bottom limit)

Therefore:

∫ 1 2 d x ( 3 − 5 x ) 2 {\displaystyle \int_{1}^{2} \frac{dx}{(3-5x)^2} } = − 1 5 ∫ − 2 − 7 d u u 2 {\displaystyle \frac{-1}{5} \int_{-2}^{-7} \frac{du}{u^2} }

Evaluating the simplified integral with respect to u u , we have:

∫ 1 2 d x ( 3 − 5 x ) 2 = 0.0714 {\displaystyle \int_{1}^{2} \frac{dx}{(3-5x)^2} } =0.0714

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