A.) ∫(2x+1)231dx
=∫(2x+1)−23dx
Let us assume that:
2x+1=u⇒2dx=du⇒dx=2du
Now:
Substituting u=2x+1 into the integration and integrating with respect to du , we have:
=21∫u−23du
=21[−21u−21]+C [ ∵∫xndx=n+1xn+1+C ( Where C is a constant of integration) ]
=−u211+C=−u1+C
Undo substitution and we have:
=−2x+11+C [ ∵u=2x+1 ]
B.) ∫sin(2x+3)dx
Let us assume that:
u=2x+3⇒du=2dx⇒dx=2du
Now:
Substituting u=2x+3 into the integration and integrating with respect to du , we have:
=21∫sin(u)du
=−21cos(u)+C [ ∵∫sin(x)dx=−cos(x)+C (Where C is a constant of integration) ]
Undo substitution and we have:
=−2cos(2x+3)+C[∵u=2x+3]
C.) ∫cosec(4x)dx
Let us assume that:
4x=u⇒4dx=du⇒dx=4du
Now:
Substituting 4x=u into the integration and integrating with respect to du , we have:
=41∫cosec(u)du
=−41ln∣cosec(u)+cot(u)∣+C [ ∵∫cosec(x)dx=−ln∣cosec(x)+cot(x)∣+C (Where C is a constant of integration)]
Undo substitution and we have:
=−4ln∣cosec(4x)+cot(4x)∣+C[∵u=4x]
D.) ∫1−9x21dx
=∫9(91−x2)1dx
=31∫(31)2−x21dx
=31sin−1(31x)+C [ ∵∫a2−x21dx=sin−1(ax)+C (Where C is a constant of integration)]
=31sin−1(3x)+C
E.) ∫1+4x21dx
=41∫41+x21dx
=41∫(21)2+x21dx
=41[211tan−1(21x)]+C [ ∵∫x2+a21dx=a1tan−1(ax)+C (Where C is a constant of integration) ]
=2tan−1(2x)+C
Comments
Dear rachael, please use the panel for submitting new questions.
∫_1^3▒〖( 3x^2 +48)dx
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