Fundamental Theorem of Calculus — Practice Problems and Worked Solutions

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16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Differentiate using the Fundamental Theorem of Calculus.

   $$\frac{d}{dx}{\int }_0^x{t}^{2}dt$$

   Answer: $x^2$

   Step-by-step solution

   1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
   2. $$f(x)\cdot \frac{d}{dx}(x)=x^2$$
2. 2.

   Differentiate using the Fundamental Theorem of Calculus.

   $$\frac{d}{dx}{\int }_0^x\sin \left(t\right)dt$$

   Answer: $\sin \left(x\right)$

   Step-by-step solution

   1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
   2. $$f(x)\cdot \frac{d}{dx}(x)=\sin \left(x\right)$$
3. 3.

   Differentiate using the Fundamental Theorem of Calculus.

   $$\frac{d}{dx}{\int }_0^{2x}{e}^{t}dt$$

   Answer: $2e^{2x}$

   Step-by-step solution

   1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
   2. $$f(2x)\cdot \frac{d}{dx}(2x)=2e^{2x}$$
4. 4.

   Differentiate using the Fundamental Theorem of Calculus.

   $$\frac{d}{dx}{\int }_0^{x^2}{t}^3+1dt$$

   Answer: $2x\left(x^6+1\right)$

   Step-by-step solution

   1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
   2. $$f(x^2)\cdot \frac{d}{dx}(x^2)=2x\left(x^6+1\right)$$
5. 5.

   Differentiate using the Fundamental Theorem of Calculus.

   $$\frac{d}{dx}{\int }_0^{3x-1}\cos \left(t\right)dt$$

   Answer: $3\cos \left(3x-1\right)$

   Step-by-step solution

   1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
   2. $$f(3x-1)\cdot \frac{d}{dx}(3x-1)=3\cos \left(3x-1\right)$$
6. 6.

   Differentiate using the Fundamental Theorem of Calculus.

   $$\frac{d}{dx}{\int }_0^{x^2+1}\frac{1}{t^2+1}dt$$

   Answer: $\frac{2x}{{\left(x^2+1\right)}^2+1}$

   Step-by-step solution

   1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
   2. $$f(x^2+1)\cdot \frac{d}{dx}(x^2+1)=\frac{2x}{{\left(x^2+1\right)}^2+1}$$
7. 7.

   Differentiate using the Fundamental Theorem of Calculus.

   $$\frac{d}{dx}{\int }_0^x{t}^{3}dt$$

   Answer: $x^3$

   Step-by-step solution

   1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
   2. $$f(x)\cdot \frac{d}{dx}(x)=x^3$$
8. 8.

   Differentiate using the Fundamental Theorem of Calculus.

   $$\frac{d}{dx}{\int }_0^x\cos \left(t\right)dt$$

   Answer: $\cos \left(x\right)$

   Step-by-step solution

   1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
   2. $$f(x)\cdot \frac{d}{dx}(x)=\cos \left(x\right)$$
9. 9.

   Differentiate using the Fundamental Theorem of Calculus.

   $$\frac{d}{dx}{\int }_0^x{t}^2+tdt$$

   Answer: $x\left(x+1\right)$

   Step-by-step solution

   1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
   2. $$f(x)\cdot \frac{d}{dx}(x)=x\left(x+1\right)$$
10. 10.

    Differentiate using the Fundamental Theorem of Calculus.

    $$\frac{d}{dx}{\int }_0^{3x}{e}^{t}dt$$

    Answer: $3e^{3x}$

    Step-by-step solution

    1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
    2. $$f(3x)\cdot \frac{d}{dx}(3x)=3e^{3x}$$
11. 11.

    Differentiate using the Fundamental Theorem of Calculus.

    $$\frac{d}{dx}{\int }_0^{x^2}{t}^{2}dt$$

    Answer: $2x^5$

    Step-by-step solution

    1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
    2. $$f(x^2)\cdot \frac{d}{dx}(x^2)=2x^5$$
12. 12.

    Differentiate using the Fundamental Theorem of Calculus.

    $$\frac{d}{dx}{\int }_0^{2x}\sin \left(t\right)dt$$

    Answer: $2\sin \left(2x\right)$

    Step-by-step solution

    1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
    2. $$f(2x)\cdot \frac{d}{dx}(2x)=2\sin \left(2x\right)$$
13. 13.

    Differentiate using the Fundamental Theorem of Calculus.

    $$\frac{d}{dx}{\int }_0^{x^3}t+1dt$$

    Answer: $3x^2\left(x^3+1\right)$

    Step-by-step solution

    1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
    2. $$f(x^3)\cdot \frac{d}{dx}(x^3)=3x^2\left(x^3+1\right)$$
14. 14.

    Differentiate using the Fundamental Theorem of Calculus.

    $$\frac{d}{dx}{\int }_0^{x^2}{e}^{t}dt$$

    Answer: $2x{e}^{x^2}$

    Step-by-step solution

    1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
    2. $$f(x^2)\cdot \frac{d}{dx}(x^2)=2x{e}^{x^2}$$
15. 15.

    Differentiate using the Fundamental Theorem of Calculus.

    $$\frac{d}{dx}{\int }_0^{2x-1}{t}^{3}dt$$

    Answer: $2{\left(2x-1\right)}^3$

    Step-by-step solution

    1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
    2. $$f(2x-1)\cdot \frac{d}{dx}(2x-1)=2{\left(2x-1\right)}^3$$
16. 16.

    Differentiate using the Fundamental Theorem of Calculus.

    $$\frac{d}{dx}{\int }_0^{x^2+2}\cos \left(t\right)dt$$

    Answer: $2x\cos \left(x^2+2\right)$

    Step-by-step solution

    1. $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))g^{\prime }(x)$$
    2. $$f(x^2+2)\cdot \frac{d}{dx}(x^2+2)=2x\cos \left(x^2+2\right)$$

More from Calculus I

• Evaluating Limits
• One-Sided Limits
• Limits at Infinity
• Continuity
• Derivative from First Principles
• Power Rule
• Trigonometric Derivatives
• Exponential & Log Derivatives
• Product Rule
• Quotient Rule
• Chain Rule
• Implicit Differentiation
• Related Rates
• Curve Analysis
• Optimization
• Antiderivatives
• Definite Integrals
• u-Substitution
• Area Between Curves
• Volumes of Revolution