10. Integration

Answer: $\frac{2 x^{5}}{5} + 3 x$

Raise each exponent by one and divide by the new exponent: $\dfrac{2x^{5}}{5} + 3 x + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{2 x^{5}}{5} + 3 x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $8 x^{3}$
• Raised the exponent but did not divide by the new one → $2 x^{5} + 3 x$

Answer: $\frac{416}{3}$

Integrate first: $\int 2 x^{2}\,dx = \frac{2 x^{3}}{3}$. Then substitute the upper limit and subtract the lower: $\left[\frac{2 x^{3}}{3}\right]_{2}^{6} = 144 - \frac{16}{3} = \frac{416}{3}$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $- \frac{416}{3}$
• Substituted the limits into the ORIGINAL function, not its integral → $64$

Answer: $\frac{3 x^{5}}{5}$

Raise each exponent by one and divide by the new exponent: $\dfrac{3x^{5}}{5} + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{3 x^{5}}{5}$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $12 x^{3}$
• Raised the exponent but did not divide by the new one → $3 x^{5}$

Answer: $4$

Integrate first: $\int x\,dx = \frac{x^{2}}{2}$. Then substitute the upper limit and subtract the lower: $\left[\frac{x^{2}}{2}\right]_{1}^{3} = \frac{9}{2} - \frac{1}{2} = 4$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $-4$
• Substituted the limits into the ORIGINAL function, not its integral → $2$

Answer: $\frac{2 x^{5}}{5} + x$

Raise each exponent by one and divide by the new exponent: $\dfrac{2x^{5}}{5} + x + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{2 x^{5}}{5} + x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $8 x^{3}$
• Raised the exponent but did not divide by the new one → $2 x^{5} + x$

Answer: $300$

Integrate first: $\int 5 x^{3}\,dx = \frac{5 x^{4}}{4}$. Then substitute the upper limit and subtract the lower: $\left[\frac{5 x^{4}}{4}\right]_{2}^{4} = 320 - 20 = 300$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $-300$
• Substituted the limits into the ORIGINAL function, not its integral → $280$

Answer: $\frac{2 x^{6}}{3} + 5 x$

Raise each exponent by one and divide by the new exponent: $\dfrac{4x^{6}}{6} + 5 x + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{2 x^{6}}{3} + 5 x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $20 x^{4}$
• Raised the exponent but did not divide by the new one → $4 x^{6} + 5 x$

Answer: $175$

Integrate first: $\int 4 x^{3}\,dx = x^{4}$. Then substitute the upper limit and subtract the lower: $\left[x^{4}\right]_{3}^{4} = 256 - 81 = 175$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $-175$
• Substituted the limits into the ORIGINAL function, not its integral → $148$

Answer: $\frac{5 x^{6}}{6} - 3 x$

Raise each exponent by one and divide by the new exponent: $\dfrac{5x^{6}}{6} - 3 x + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{5 x^{6}}{6} - 3 x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $25 x^{4}$
• Raised the exponent but did not divide by the new one → $5 x^{6} - 3 x$

Answer: $\frac{45}{2}$

Integrate first: $\int 5 x\,dx = \frac{5 x^{2}}{2}$. Then substitute the upper limit and subtract the lower: $\left[\frac{5 x^{2}}{2}\right]_{0}^{3} = \frac{45}{2} - 0 = \frac{45}{2}$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $- \frac{45}{2}$
• Substituted the limits into the ORIGINAL function, not its integral → $15$

Answer: $\frac{x^{6}}{2} - 7 x$

Raise each exponent by one and divide by the new exponent: $\dfrac{3x^{6}}{6} - 7 x + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{x^{6}}{2} - 7 x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $15 x^{4}$
• Raised the exponent but did not divide by the new one → $3 x^{6} - 7 x$

Answer: $316$

Integrate first: $\int 3 x^{2}\,dx = x^{3}$. Then substitute the upper limit and subtract the lower: $\left[x^{3}\right]_{3}^{7} = 343 - 27 = 316$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $-316$
• Substituted the limits into the ORIGINAL function, not its integral → $120$

Answer: $\frac{x^{3}}{3} - 7 x$

Raise each exponent by one and divide by the new exponent: $\dfrac{1x^{3}}{3} - 7 x + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{x^{3}}{3} - 7 x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $2 x$
• Raised the exponent but did not divide by the new one → $x^{3} - 7 x$

Answer: $4$

Integrate first: $\int x^{3}\,dx = \frac{x^{4}}{4}$. Then substitute the upper limit and subtract the lower: $\left[\frac{x^{4}}{4}\right]_{0}^{2} = 4 - 0 = 4$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $-4$
• Substituted the limits into the ORIGINAL function, not its integral → $8$

Answer: $\frac{3 x^{5}}{5} + x$

Raise each exponent by one and divide by the new exponent: $\dfrac{3x^{5}}{5} + x + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{3 x^{5}}{5} + x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $12 x^{3}$
• Raised the exponent but did not divide by the new one → $3 x^{5} + x$

Answer: $\frac{4}{3}$

Integrate first: $\int 4 x^{2}\,dx = \frac{4 x^{3}}{3}$. Then substitute the upper limit and subtract the lower: $\left[\frac{4 x^{3}}{3}\right]_{0}^{1} = \frac{4}{3} - 0 = \frac{4}{3}$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $- \frac{4}{3}$
• Substituted the limits into the ORIGINAL function, not its integral → $4$

Answer: $2 x^{3} + 7 x$

Raise each exponent by one and divide by the new exponent: $\dfrac{6x^{3}}{3} + 7 x + C$. Passing through the origin makes $C = 0$, so the answer is $2 x^{3} + 7 x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $12 x$
• Raised the exponent but did not divide by the new one → $6 x^{3} + 7 x$

Answer: $30$

Integrate first: $\int 5 x\,dx = \frac{5 x^{2}}{2}$. Then substitute the upper limit and subtract the lower: $\left[\frac{5 x^{2}}{2}\right]_{2}^{4} = 40 - 10 = 30$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $-30$
• Substituted the limits into the ORIGINAL function, not its integral → $10$

Answer: $\frac{4 x^{3}}{3} - 5 x$

Raise each exponent by one and divide by the new exponent: $\dfrac{4x^{3}}{3} - 5 x + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{4 x^{3}}{3} - 5 x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $8 x$
• Raised the exponent but did not divide by the new one → $4 x^{3} - 5 x$

Answer: $\frac{81}{2}$

Integrate first: $\int 2 x^{3}\,dx = \frac{x^{4}}{2}$. Then substitute the upper limit and subtract the lower: $\left[\frac{x^{4}}{2}\right]_{0}^{3} = \frac{81}{2} - 0 = \frac{81}{2}$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $- \frac{81}{2}$
• Substituted the limits into the ORIGINAL function, not its integral → $54$

Answer: $\frac{4 x^{5}}{5} - 8 x$

Raise each exponent by one and divide by the new exponent: $\dfrac{4x^{5}}{5} - 8 x + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{4 x^{5}}{5} - 8 x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $16 x^{3}$
• Raised the exponent but did not divide by the new one → $4 x^{5} - 8 x$

Answer: $\frac{75}{4}$

Integrate first: $\int 5 x^{3}\,dx = \frac{5 x^{4}}{4}$. Then substitute the upper limit and subtract the lower: $\left[\frac{5 x^{4}}{4}\right]_{1}^{2} = 20 - \frac{5}{4} = \frac{75}{4}$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $- \frac{75}{4}$
• Substituted the limits into the ORIGINAL function, not its integral → $35$

Answer: $\frac{2 x^{5}}{5} - 2 x$

Raise each exponent by one and divide by the new exponent: $\dfrac{2x^{5}}{5} - 2 x + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{2 x^{5}}{5} - 2 x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $8 x^{3}$
• Raised the exponent but did not divide by the new one → $2 x^{5} - 2 x$

Answer: $80$

Integrate first: $\int 4 x\,dx = 2 x^{2}$. Then substitute the upper limit and subtract the lower: $\left[2 x^{2}\right]_{3}^{7} = 98 - 18 = 80$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $-80$
• Substituted the limits into the ORIGINAL function, not its integral → $16$

Answer: $\frac{2 x^{3}}{3}$

Raise each exponent by one and divide by the new exponent: $\dfrac{2x^{3}}{3} + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{2 x^{3}}{3}$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $4 x$
• Raised the exponent but did not divide by the new one → $2 x^{3}$

Answer: $\frac{1275}{4}$

Integrate first: $\int 5 x^{3}\,dx = \frac{5 x^{4}}{4}$. Then substitute the upper limit and subtract the lower: $\left[\frac{5 x^{4}}{4}\right]_{1}^{4} = 320 - \frac{5}{4} = \frac{1275}{4}$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $- \frac{1275}{4}$
• Substituted the limits into the ORIGINAL function, not its integral → $315$

Answer: $\frac{6 x^{5}}{5} + 8 x$

Raise each exponent by one and divide by the new exponent: $\dfrac{6x^{5}}{5} + 8 x + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{6 x^{5}}{5} + 8 x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $24 x^{3}$
• Raised the exponent but did not divide by the new one → $6 x^{5} + 8 x$

Answer: $\frac{243}{4}$

Integrate first: $\int 3 x^{3}\,dx = \frac{3 x^{4}}{4}$. Then substitute the upper limit and subtract the lower: $\left[\frac{3 x^{4}}{4}\right]_{0}^{3} = \frac{243}{4} - 0 = \frac{243}{4}$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $- \frac{243}{4}$
• Substituted the limits into the ORIGINAL function, not its integral → $81$

Answer: $\frac{x^{6}}{6} + x$

Raise each exponent by one and divide by the new exponent: $\dfrac{1x^{6}}{6} + x + C$. Passing through the origin makes $C = 0$, so the answer is $\frac{x^{6}}{6} + x$.

Common mistakes, and the answer each one gives:

• Differentiated instead of integrating → $5 x^{4}$
• Raised the exponent but did not divide by the new one → $x^{6} + x$

Answer: $\frac{21}{2}$

Integrate first: $\int x\,dx = \frac{x^{2}}{2}$. Then substitute the upper limit and subtract the lower: $\left[\frac{x^{2}}{2}\right]_{2}^{5} = \frac{25}{2} - 2 = \frac{21}{2}$.

Common mistakes, and the answer each one gives:

• Subtracted the wrong way round → $- \frac{21}{2}$
• Substituted the limits into the ORIGINAL function, not its integral → $3$