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GRE Integrals and Applications — practice questions — page 2

Explore 44 free Integrals and Applications questions in GRE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Integrals and Applications questions and exercises with answers

Showing questions 31–44 of 44.

$\displaystyle\int x^n\,dx$ (where $n \ne -1$) equals:Read the answer and worked solution →The integral ∫ x^n dx (for n ≠ -1) equals:Read the answer and worked solution →Using integration by parts, $\displaystyle\int x e^x\,dx$ equals:Read the answer and worked solution →To solve the homogeneous differential equation $\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{xy}$, the appropriate substitution is:Read the answer and worked solution →Using partial fractions, $\displaystyle\int \dfrac{1}{(x-1)(x-2)}\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \dfrac{1}{x^2 - a^2}\,dx$ (with $|x| > a > 0$) equals:Read the answer and worked solution →$\displaystyle\int \dfrac{1}{1+x^2}\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \dfrac{1}{\sqrt{1 - x^2}}\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \sec^2 x\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \log x\,dx$ equals:Read the answer and worked solution →$\displaystyle\int e^x\bigl(f(x) + f'(x)\bigr)\,dx$ equals:Read the answer and worked solution →Using the substitution $t = 1 + x^2$, $\displaystyle\int \dfrac{x}{1 + x^2}\,dx$ equals:Read the answer and worked solution →The number of arbitrary constants in the general solution of a differential equation of order $n$ is:Read the answer and worked solution →The integrating factor of the linear differential equation $\dfrac{dy}{dx} + \dfrac{y}{x} = x^2$ is:Read the answer and worked solution →

Worked practice questions

$\displaystyle\int x^n\,dx$ (where $n \ne -1$) equals:

  1. $x^n \log x + C$
  2. $\dfrac{x^{n-1}}{n-1} + C$
  3. $\dfrac{x^{n+1}}{n+1} + C$
  4. $n x^{n-1} + C$
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Correct answer: C. $\dfrac{x^{n+1}}{n+1} + C$

$\dfrac{d}{dx}\left(\dfrac{x^{n+1}}{n+1}\right) = x^n$ for $n \ne -1$ so by anti-differentiation $\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$. The exception $n = -1$ is handled separately as $\int \dfrac{1}{x}\,dx = \log|x| + C$.

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The integral ∫ x^n dx (for n ≠ -1) equals:

  1. $n x^{n-1} + C$ (derivative formula)
  2. $x^{n+1} + C$ (missing divisor)
  3. $x^n + C$ (no change)
  4. $x^{n+1} / (n+1) + C$ (power rule)
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Correct answer: D. $x^{n+1} / (n+1) + C$ (power rule)

Power rule for integration: ∫x^n dx = x^(n+1)/(n+1) + C, valid for n ≠ -1. For n = -1, ∫(1/x)dx = ln|x| + C.

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Using integration by parts, $\displaystyle\int x e^x\,dx$ equals:

  1. $(x + 1) e^x + C$
  2. $\dfrac{x^2}{2} e^x + C$
  3. $(x - 1) e^x + C$
  4. $x e^x + C$
Show answer and solution

Correct answer: C. $(x - 1) e^x + C$

Take $u = x$ (algebraic) and $dv = e^x\,dx$, then $du = dx$ and $v = e^x$. **By parts:** $\int x e^x\,dx = x e^x - \int e^x\,dx = x e^x - e^x + C = (x - 1) e^x + C$.

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More Mathematics topics

MathematicsStatistics and ProbabilityArithmetic ProgressionSets and RelationsQuadratic EquationsReal Numbers and PolynomialsUnit 1A1: Relations, Functions and GraphsUnit 6B: Definite Integration & Riemann Sums