Write the integral in terms of u and du. Then evaluate: ∫(x+25)−2dx,u=x+25

Question

George Spencer · Accepted Answer

Step 1
In some cases integrals can be transformed as a standard integral using appropriate substitution. For example for the integral

\int \tan (x) \sec^{2} (x) d x

the substitution

\tan (x) = u

transforms the integral to

\int u d u

which can be easily integrated.
For the given case need to write the given integral in terms of the substituted variable. Use the substitution to write an appropriate differential and substitute in the integral.
Step 2
Given integral is

\int {(x + 25)}^{- 2} d x

. The substitution to be used is u=x+25. Differentiate this to get du=dx. Substitute this in the integral and integrate.

\int {(x + 25)}^{- 2} d x = \int u^{- 2} d u

= \frac{u^{- 2 + 1}}{- 2 + 1} + C

= - \frac{1}{u} + C

= - \frac{1}{x + 25} + C

Gloria Lusk · Accepted Answer

Step 1: Remove parentheses.  ∫(x+25)−2dx  Step 2: Use Negative Power Rule: x−a=1xa.  ∫1(x+25)2dx  Step 3: Use Integration by Substitution.  Let u=x+25, du=dx  Step 4: Using u and du above, rewrite ∫1(x+25)2dx.  ∫1u2du  Step 5: Use Power Rule: ∫xndx=xn+1n+1+C.  −1u  Step 6: Substitute u=x+25 back into the original integral.  −1x+25  Step 7: Add constant.  −1x+25+C

Write the integral in terms of u and du. Then

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