Evaluate the integral \int ((t+1)^{2}-\frac{1}{t^{4}})dt

katelineliseua 2021-11-22 Answered
Evaluate the integral \(\displaystyle\int{\left({\left({t}+{1}\right)}^{{{2}}}-{\frac{{{1}}}{{{t}^{{{4}}}}}}\right)}{\left.{d}{t}\right.}\)

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Expert Answer

Vaing1990
Answered 2021-11-23 Author has 1722 answers
Step 1
To evaluate the integral: \(\displaystyle\int{\left({\left({t}+{1}\right)}^{{{2}}}-{\frac{{{1}}}{{{t}^{{{4}}}}}}\right)}{\left.{d}{t}\right.}\)
Evaluating the given integral.
\(\displaystyle\int{\left({\left({t}+{1}\right)}^{{{2}}}-{\frac{{{1}}}{{{t}^{{{4}}}}}}\right)}{\left.{d}{t}\right.}=\int{\left[{\left({t}^{{{2}}}+{2}{t}+{1}\right)}-{\frac{{{1}}}{{{t}^{{{4}}}}}}\right]}{\left.{d}{t}\right.}\)
\(\displaystyle=\int{t}^{{{2}}}{\left.{d}{t}\right.}+{2}\int{t}{\left.{d}{t}\right.}+\int{\left.{d}{t}\right.}-\int{\frac{{{1}}}{{{t}^{{{4}}}}}}{\left.{d}{t}\right.}\)
\(\displaystyle={\frac{{{t}^{{{3}}}}}{{{3}}}}+{2}\cdot{\frac{{{t}^{{{2}}}}}{{{2}}}}+{t}+{3}{\frac{{{1}}}{{{t}^{{{3}}}}}}+{C}\)
\(\displaystyle={\frac{{{t}^{{{3}}}}}{{{3}}}}+{t}^{{{2}}}+{t}+{\frac{{{3}}}{{{t}^{{{3}}}}}}+{C}\)
Step 2
Hence, required answer is \(\displaystyle{\left[{\frac{{{t}^{{{3}}}}}{{{3}}}}+{t}^{{{2}}}+{t}+{\frac{{{3}}}{{{t}^{{{3}}}}}}\right]}+{C}\)
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Gloria Lusk
Answered 2021-11-24 Author has 759 answers
Step 1: Expand.
\(\displaystyle\int{\left({t}+{1}\right)}^{{{2}}}-{\frac{{{1}}}{{{t}^{{{4}}}}}}{\left.{d}{t}\right.}\)
Step 2: Use Sum Rule: \(\displaystyle\int{f{{\left({x}\right)}}}+{g{{\left({x}\right)}}}{\left.{d}{x}\right.}=\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}+\int{g{{\left({x}\right)}}}{\left.{d}{x}\right.}\).
\(\displaystyle\int{\left({t}+{1}\right)}^{{{2}}}{\left.{d}{t}\right.}-\int{\frac{{{1}}}{{{t}^{{{4}}}}}}{\left.{d}{t}\right.}\)
Step 3: Expand.
\(\displaystyle\int{t}^{{{2}}}+{2}{t}+{1}{\left.{d}{t}\right.}-\int{\frac{{{1}}}{{{t}^{{{4}}}}}}{\left.{d}{t}\right.}\)
Step 4: Use Power Rule: \(\displaystyle\int{x}^{{{n}}}{\left.{d}{x}\right.}={\frac{{{x}^{{{n}+{1}}}}}{{{n}+{1}}}}+{C}\).
\(\displaystyle{\frac{{{t}^{{{3}}}}}{{{3}}}}+{t}^{{{2}}}+{t}-\int{\frac{{{1}}}{{{t}^{{{4}}}}}}{\left.{d}{t}\right.}\)
Step 5: Use Power Rule: \(\displaystyle\int{x}^{{{n}}}{\left.{d}{x}\right.}={\frac{{{x}^{{{n}+{1}}}}}{{{n}+{1}}}}+{C}\).
\(\displaystyle{\frac{{{t}^{{{3}}}}}{{{3}}}}+{t}^{{{2}}}+{t}+{\frac{{{1}}}{{{3}{t}^{{{3}}}}}}\)
Step 6: Add constant.
\(\displaystyle{\frac{{{t}^{{{3}}}}}{{{3}}}}+{t}^{{{2}}}+{t}+{\frac{{{1}}}{{{3}{t}^{{{3}}}}}}+{C}\)
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