Answer to Question #167019 in Calculus for Angelo

Question #167019

Evaluate the following indefinite integrals:

  1. ∫ (12x^5 − 6𝑥^3 − 4x + 1/2 ) dx
  2. ∫ (√𝑥 + √3)^2 dx
  3. ∫ (𝑥^𝑒 − 𝑥/𝑒 + 2𝑒𝑥) dx
  4. ∫ (2𝑡+1)(2𝑡−1)/2√𝑡 dx
  5. ∫ 8𝑦^3 −1/2𝑦+1 dx
1
Expert's answer
2021-02-28T17:04:57-0500

Ques:1:

\intop (12x56x34x+12)dx(12x^5-6x^3-4x+\dfrac{1}{2})dx

Distribute the integeration on each term

12x5dx6X3dx4xdx+12dx\Rightarrow\smallint12x^5dx-\smallint6X^3dx-\smallint4xdx+\smallint\dfrac{1}{2}dx

12×x(5+1)5+16×x(3+1)3+14×x1+11+1+12×x0+10+1\Rightarrow12\times\dfrac{x^{(5+1)}}{5+1}-6\times\dfrac{x^{(3+1)}}{3+1}-4\times\dfrac{x^{1+1}}{1+1}+\dfrac{1}{2}\times\dfrac{x^{0+1}}{0+1}


\Rightarrow 12×x666×x444×x22+12×x12\times\dfrac{x^6}{6}-6\times\dfrac{x^4}{4}-4\times\dfrac{x^2}{2}+\dfrac{1}{2}\times x


2x632x42x2+x2\Rightarrow2x^6-\dfrac{3}{2}x^4-2x^2+\dfrac{x}{2}



Ques :2:

(x+3)2dx\smallint(\sqrt{x}+\sqrt{3})^2dx


(x+3+23x)dx\Rightarrow\smallint( x+3+2\sqrt{3}\sqrt{x})dx


xdx+3dx+23×x12dx\Rightarrow \smallint xdx+3\smallint dx+\smallint2\sqrt{3}\times x^\dfrac{1}{2}dx


x22+3x+23×x12+112+1\Rightarrow \dfrac{x^2}{2}+3x+2\sqrt{3}\times\dfrac{x^{\dfrac{1}{2}+1}}{\dfrac{1}{2}+1}


x22+3x+23×x3232\Rightarrow\dfrac{x^2}{2}+3x+2\sqrt{3}\times \dfrac{x^{\dfrac{3}{2}}}{\dfrac{3}{2}}


Ques::3::


(xexe+2×e×x)dx\smallint( x^e-\dfrac{x}{e}+2\times e \times x)dx


xedxxedx+2exdx\Rightarrow\smallint x^edx-\smallint \dfrac{x}{e}dx+\smallint2exdx


xe+1e+1x22e+ex2\Rightarrow \dfrac{x^{e+1}}{e+1}-\dfrac{x^2}{2e}+ex^2


Ques::4::


(2t+1)(2t1)2tdx\smallint\dfrac{(2t+1)(2t-1)}{2\sqrt{t}}dx


As we have to integerate in terms of xx .so the terms containing tt can be taken outside the integeration


\Rightarrow (2t+1)(2t1)2tdx\dfrac{(2t+1)(2t-1)}{2\sqrt{t}}\smallint dx


(4t21)2t×x\Rightarrow\dfrac{(4t^2-1)}{2\sqrt{t}}\times x


Ques::5::


(8y312y+1)dx\smallint(8y^3-\dfrac{1}{2y}+1)dx


8y3dx12ydx+dx\Rightarrow8y^3\smallint dx- \dfrac{1}{2y}\smallint dx+\smallint dx


8y3×xx2y+x\Rightarrow 8y^3\times x-\dfrac{x}{2y}+x







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